Runtime Complexity (innermost) proof of /tmp/tmpqCvzx2/MYNAT_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), 20 ms)

↳2 CpxTRS

    ↳3 CpxTrsMatchBoundsTAProof (⇔, 573 ms)

    ↳4 BOUNDS(1, n^1)

    ↳5 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 10 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)), 226 ms)

        ↳14 CdtProblem

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

        ↳16 CdtProblem

        ↳17 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 96 ms)

        ↳18 CdtProblem

        ↳19 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 73 ms)

        ↳20 CdtProblem

        ↳21 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 54 ms)

        ↳22 CdtProblem

        ↳23 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 61 ms)

        ↳24 CdtProblem

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

        ↳26 CdtProblem

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

        ↳28 CdtProblem

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

        ↳30 CdtProblem

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

        ↳32 CdtProblem

        ↳33 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 12 ms)

        ↳34 CdtProblem

        ↳35 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 64 ms)

        ↳36 CdtProblem

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

        ↳38 CdtProblem

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

        ↳40 CdtProblem

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

        ↳42 CdtProblem

        ↳43 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 60 ms)

        ↳44 CdtProblem

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

        ↳46 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(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(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(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))

(2) Obligation:

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

The TRS R consists of the following rules:

top(ok(X)) → top(active(X))
U33(ok(X)) → ok(U33(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U33(mark(X)) → mark(U33(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U61(ok(X)) → ok(U61(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U41(mark(X1), X2) → mark(U41(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
proper(tt) → ok(tt)
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U13(ok(X)) → ok(U13(X))
U13(mark(X)) → mark(U13(X))
U22(mark(X)) → mark(U22(X))
x(mark(X1), X2) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
U22(ok(X)) → ok(U22(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
s(ok(X)) → ok(s(X))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(0) → ok(0)
top(mark(X)) → top(proper(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:
ok0(0) → 0
active0(0) → 0
mark0(0) → 0
tt0() → 0
00() → 0
top0(0) → 1
U330(0) → 2
U310(0, 0, 0) → 3
U320(0, 0) → 4
isNat0(0) → 5
U710(0, 0, 0) → 6
U610(0) → 7
and0(0, 0) → 8
U510(0, 0, 0) → 9
U410(0, 0) → 10
plus0(0, 0) → 11
U210(0, 0) → 12
proper0(0) → 13
isNatKind0(0) → 14
U110(0, 0, 0) → 15
U130(0) → 16
U220(0) → 17
x0(0, 0) → 18
U120(0, 0) → 19
s0(0) → 20
active1(0) → 21
top1(21) → 1
U331(0) → 22
ok1(22) → 2
U311(0, 0, 0) → 23
ok1(23) → 3
U321(0, 0) → 24
mark1(24) → 4
isNat1(0) → 25
ok1(25) → 5
U711(0, 0, 0) → 26
ok1(26) → 6
U331(0) → 27
mark1(27) → 2
U611(0) → 28
mark1(28) → 7
U711(0, 0, 0) → 29
mark1(29) → 6
U611(0) → 30
ok1(30) → 7
and1(0, 0) → 31
ok1(31) → 8
U511(0, 0, 0) → 32
mark1(32) → 9
U411(0, 0) → 33
mark1(33) → 10
U321(0, 0) → 34
ok1(34) → 4
plus1(0, 0) → 35
ok1(35) → 11
plus1(0, 0) → 36
mark1(36) → 11
U211(0, 0) → 37
mark1(37) → 12
U311(0, 0, 0) → 38
mark1(38) → 3
tt1() → 39
ok1(39) → 13
U511(0, 0, 0) → 40
ok1(40) → 9
isNatKind1(0) → 41
ok1(41) → 14
U111(0, 0, 0) → 42
mark1(42) → 15
U131(0) → 43
ok1(43) → 16
U131(0) → 44
mark1(44) → 16
U221(0) → 45
mark1(45) → 17
x1(0, 0) → 46
mark1(46) → 18
and1(0, 0) → 47
mark1(47) → 8
U121(0, 0) → 48
ok1(48) → 19
U221(0) → 49
ok1(49) → 17
U111(0, 0, 0) → 50
ok1(50) → 15
U411(0, 0) → 51
ok1(51) → 10
U121(0, 0) → 52
mark1(52) → 19
x1(0, 0) → 53
ok1(53) → 18
U211(0, 0) → 54
ok1(54) → 12
s1(0) → 55
ok1(55) → 20
s1(0) → 56
mark1(56) → 20
01() → 57
ok1(57) → 13
proper1(0) → 58
top1(58) → 1
ok1(22) → 22
ok1(22) → 27
ok1(23) → 23
ok1(23) → 38
mark1(24) → 24
mark1(24) → 34
ok1(25) → 25
ok1(26) → 26
ok1(26) → 29
mark1(27) → 22
mark1(27) → 27
mark1(28) → 28
mark1(28) → 30
mark1(29) → 26
mark1(29) → 29
ok1(30) → 28
ok1(30) → 30
ok1(31) → 31
ok1(31) → 47
mark1(32) → 32
mark1(32) → 40
mark1(33) → 33
mark1(33) → 51
ok1(34) → 24
ok1(34) → 34
ok1(35) → 35
ok1(35) → 36
mark1(36) → 35
mark1(36) → 36
mark1(37) → 37
mark1(37) → 54
mark1(38) → 23
mark1(38) → 38
ok1(39) → 58
ok1(40) → 32
ok1(40) → 40
ok1(41) → 41
mark1(42) → 42
mark1(42) → 50
ok1(43) → 43
ok1(43) → 44
mark1(44) → 43
mark1(44) → 44
mark1(45) → 45
mark1(45) → 49
mark1(46) → 46
mark1(46) → 53
mark1(47) → 31
mark1(47) → 47
ok1(48) → 48
ok1(48) → 52
ok1(49) → 45
ok1(49) → 49
ok1(50) → 42
ok1(50) → 50
ok1(51) → 33
ok1(51) → 51
mark1(52) → 48
mark1(52) → 52
ok1(53) → 46
ok1(53) → 53
ok1(54) → 37
ok1(54) → 54
ok1(55) → 55
ok1(55) → 56
mark1(56) → 55
mark1(56) → 56
ok1(57) → 58
active2(39) → 59
top2(59) → 1
active2(57) → 59

(4) BOUNDS(1, n^1)

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

Converted Cpx (relative) TRS to CDT

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
U33(ok(z0)) → ok(U33(z0))
U33(mark(z0)) → mark(U33(z0))
U31(ok(z0), ok(z1), ok(z2)) → ok(U31(z0, z1, z2))
U31(mark(z0), z1, z2) → mark(U31(z0, z1, z2))
U32(mark(z0), z1) → mark(U32(z0, z1))
U32(ok(z0), ok(z1)) → ok(U32(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U71(ok(z0), ok(z1), ok(z2)) → ok(U71(z0, z1, z2))
U71(mark(z0), z1, z2) → mark(U71(z0, z1, z2))
U61(mark(z0)) → mark(U61(z0))
U61(ok(z0)) → ok(U61(z0))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
U51(mark(z0), z1, z2) → mark(U51(z0, z1, z2))
U51(ok(z0), ok(z1), ok(z2)) → ok(U51(z0, z1, z2))
U41(mark(z0), z1) → mark(U41(z0, z1))
U41(ok(z0), ok(z1)) → ok(U41(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(mark(z0), z1) → mark(plus(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
proper(tt) → ok(tt)
proper(0) → ok(0)
isNatKind(ok(z0)) → ok(isNatKind(z0))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
U13(ok(z0)) → ok(U13(z0))
U13(mark(z0)) → mark(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
x(mark(z0), z1) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
U12(mark(z0), z1) → mark(U12(z0, z1))
s(ok(z0)) → ok(s(z0))
s(mark(z0)) → mark(s(z0))

Tuples:

TOP(ok(z0)) → c(TOP(active(z0)))
TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))
U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
U61'(mark(z0)) → c11(U61'(z0))
U61'(ok(z0)) → c12(U61'(z0))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0), z1) → c22(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
PROPER(tt) → c24
PROPER(0) → c25
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
X(mark(z0), z1) → c33(X(z0, z1))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))

S tuples:

TOP(ok(z0)) → c(TOP(active(z0)))
TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))
U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
U61'(mark(z0)) → c11(U61'(z0))
U61'(ok(z0)) → c12(U61'(z0))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0), z1) → c22(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
PROPER(tt) → c24
PROPER(0) → c25
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
X(mark(z0), z1) → c33(X(z0, z1))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))

K tuples:none
Defined Rule Symbols:

top, U33, U31, U32, isNat, U71, U61, and, U51, U41, plus, U21, proper, isNatKind, U11, U13, U22, x, U12, s

Defined Pair Symbols:

TOP, U33', U31', U32', ISNAT, U71', U61', AND, U51', U41', PLUS, U21', PROPER, ISNATKIND, U11', U13', U22', X, U12', S

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

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

Removed 3 trailing nodes:

PROPER(0) → c25
PROPER(tt) → c24
TOP(ok(z0)) → c(TOP(active(z0)))

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
U33(ok(z0)) → ok(U33(z0))
U33(mark(z0)) → mark(U33(z0))
U31(ok(z0), ok(z1), ok(z2)) → ok(U31(z0, z1, z2))
U31(mark(z0), z1, z2) → mark(U31(z0, z1, z2))
U32(mark(z0), z1) → mark(U32(z0, z1))
U32(ok(z0), ok(z1)) → ok(U32(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U71(ok(z0), ok(z1), ok(z2)) → ok(U71(z0, z1, z2))
U71(mark(z0), z1, z2) → mark(U71(z0, z1, z2))
U61(mark(z0)) → mark(U61(z0))
U61(ok(z0)) → ok(U61(z0))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
U51(mark(z0), z1, z2) → mark(U51(z0, z1, z2))
U51(ok(z0), ok(z1), ok(z2)) → ok(U51(z0, z1, z2))
U41(mark(z0), z1) → mark(U41(z0, z1))
U41(ok(z0), ok(z1)) → ok(U41(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(mark(z0), z1) → mark(plus(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
proper(tt) → ok(tt)
proper(0) → ok(0)
isNatKind(ok(z0)) → ok(isNatKind(z0))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
U13(ok(z0)) → ok(U13(z0))
U13(mark(z0)) → mark(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
x(mark(z0), z1) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
U12(mark(z0), z1) → mark(U12(z0, z1))
s(ok(z0)) → ok(s(z0))
s(mark(z0)) → mark(s(z0))

Tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))
U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
U61'(mark(z0)) → c11(U61'(z0))
U61'(ok(z0)) → c12(U61'(z0))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0), z1) → c22(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
X(mark(z0), z1) → c33(X(z0, z1))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))

S tuples:

TOP(mark(z0)) → c1(TOP(proper(z0)), PROPER(z0))
U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
U61'(mark(z0)) → c11(U61'(z0))
U61'(ok(z0)) → c12(U61'(z0))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0), z1) → c22(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
X(mark(z0), z1) → c33(X(z0, z1))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))

K tuples:none
Defined Rule Symbols:

top, U33, U31, U32, isNat, U71, U61, and, U51, U41, plus, U21, proper, isNatKind, U11, U13, U22, x, U12, s

Defined Pair Symbols:

TOP, U33', U31', U32', ISNAT, U71', U61', AND, U51', U41', PLUS, U21', ISNATKIND, U11', U13', U22', X, U12', S

Compound Symbols:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39

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

Removed 1 trailing tuple parts

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
U33(ok(z0)) → ok(U33(z0))
U33(mark(z0)) → mark(U33(z0))
U31(ok(z0), ok(z1), ok(z2)) → ok(U31(z0, z1, z2))
U31(mark(z0), z1, z2) → mark(U31(z0, z1, z2))
U32(mark(z0), z1) → mark(U32(z0, z1))
U32(ok(z0), ok(z1)) → ok(U32(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U71(ok(z0), ok(z1), ok(z2)) → ok(U71(z0, z1, z2))
U71(mark(z0), z1, z2) → mark(U71(z0, z1, z2))
U61(mark(z0)) → mark(U61(z0))
U61(ok(z0)) → ok(U61(z0))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
U51(mark(z0), z1, z2) → mark(U51(z0, z1, z2))
U51(ok(z0), ok(z1), ok(z2)) → ok(U51(z0, z1, z2))
U41(mark(z0), z1) → mark(U41(z0, z1))
U41(ok(z0), ok(z1)) → ok(U41(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(mark(z0), z1) → mark(plus(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
proper(tt) → ok(tt)
proper(0) → ok(0)
isNatKind(ok(z0)) → ok(isNatKind(z0))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
U13(ok(z0)) → ok(U13(z0))
U13(mark(z0)) → mark(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
x(mark(z0), z1) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
U12(mark(z0), z1) → mark(U12(z0, z1))
s(ok(z0)) → ok(s(z0))
s(mark(z0)) → mark(s(z0))

Tuples:

U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
U61'(mark(z0)) → c11(U61'(z0))
U61'(ok(z0)) → c12(U61'(z0))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0), z1) → c22(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
X(mark(z0), z1) → c33(X(z0, z1))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
U61'(mark(z0)) → c11(U61'(z0))
U61'(ok(z0)) → c12(U61'(z0))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0), z1) → c22(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
X(mark(z0), z1) → c33(X(z0, z1))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

K tuples:none
Defined Rule Symbols:

top, U33, U31, U32, isNat, U71, U61, and, U51, U41, plus, U21, proper, isNatKind, U11, U13, U22, x, U12, s

Defined Pair Symbols:

U33', U31', U32', ISNAT, U71', U61', AND, U51', U41', PLUS, U21', ISNATKIND, U11', U13', U22', X, U12', S, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c1

(11) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
U33(ok(z0)) → ok(U33(z0))
U33(mark(z0)) → mark(U33(z0))
U31(ok(z0), ok(z1), ok(z2)) → ok(U31(z0, z1, z2))
U31(mark(z0), z1, z2) → mark(U31(z0, z1, z2))
U32(mark(z0), z1) → mark(U32(z0, z1))
U32(ok(z0), ok(z1)) → ok(U32(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U71(ok(z0), ok(z1), ok(z2)) → ok(U71(z0, z1, z2))
U71(mark(z0), z1, z2) → mark(U71(z0, z1, z2))
U61(mark(z0)) → mark(U61(z0))
U61(ok(z0)) → ok(U61(z0))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
U51(mark(z0), z1, z2) → mark(U51(z0, z1, z2))
U51(ok(z0), ok(z1), ok(z2)) → ok(U51(z0, z1, z2))
U41(mark(z0), z1) → mark(U41(z0, z1))
U41(ok(z0), ok(z1)) → ok(U41(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(mark(z0), z1) → mark(plus(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
U13(ok(z0)) → ok(U13(z0))
U13(mark(z0)) → mark(U13(z0))
U22(mark(z0)) → mark(U22(z0))
U22(ok(z0)) → ok(U22(z0))
x(mark(z0), z1) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
U12(ok(z0), ok(z1)) → ok(U12(z0, z1))
U12(mark(z0), z1) → mark(U12(z0, z1))
s(ok(z0)) → ok(s(z0))
s(mark(z0)) → mark(s(z0))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)

Tuples:

U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
U61'(mark(z0)) → c11(U61'(z0))
U61'(ok(z0)) → c12(U61'(z0))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0), z1) → c22(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
X(mark(z0), z1) → c33(X(z0, z1))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
U61'(mark(z0)) → c11(U61'(z0))
U61'(ok(z0)) → c12(U61'(z0))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0), z1) → c22(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
X(mark(z0), z1) → c33(X(z0, z1))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

K tuples:none
Defined Rule Symbols:

proper

Defined Pair Symbols:

U33', U31', U32', ISNAT, U71', U61', AND, U51', U41', PLUS, U21', ISNATKIND, U11', U13', U22', X, U12', S, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c1

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

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

U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))

We considered the (Usable) Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)

And the Tuples:

U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
U61'(mark(z0)) → c11(U61'(z0))
U61'(ok(z0)) → c12(U61'(z0))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0), z1) → c22(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
X(mark(z0), z1) → c33(X(z0, z1))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

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

POL(0) = 0
POL(AND(x₁, x₂)) = 0
POL(ISNAT(x₁)) = 0
POL(ISNATKIND(x₁)) = 0
POL(PLUS(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(TOP(x₁)) = [2]x₁
POL(U11'(x₁, x₂, x₃)) = x₂ + x₃
POL(U12'(x₁, x₂)) = 0
POL(U13'(x₁)) = 0
POL(U21'(x₁, x₂)) = 0
POL(U22'(x₁)) = 0
POL(U31'(x₁, x₂, x₃)) = 0
POL(U32'(x₁, x₂)) = 0
POL(U33'(x₁)) = 0
POL(U41'(x₁, x₂)) = 0
POL(U51'(x₁, x₂, x₃)) = 0
POL(U61'(x₁)) = 0
POL(U71'(x₁, x₂, x₃)) = 0
POL(X(x₁, x₂)) = 0
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(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(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(mark(x₁)) = [2] + x₁
POL(ok(x₁)) = [1] + x₁
POL(proper(x₁)) = [2]
POL(tt) = 0

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)

Tuples:

U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
U61'(mark(z0)) → c11(U61'(z0))
U61'(ok(z0)) → c12(U61'(z0))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0), z1) → c22(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
X(mark(z0), z1) → c33(X(z0, z1))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
U61'(mark(z0)) → c11(U61'(z0))
U61'(ok(z0)) → c12(U61'(z0))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0), z1) → c22(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
X(mark(z0), z1) → c33(X(z0, z1))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

K tuples:

U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U33', U31', U32', ISNAT, U71', U61', AND, U51', U41', PLUS, U21', ISNATKIND, U11', U13', U22', X, U12', S, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c1

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

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

U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))

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

U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
U61'(mark(z0)) → c11(U61'(z0))
U61'(ok(z0)) → c12(U61'(z0))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0), z1) → c22(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
X(mark(z0), z1) → c33(X(z0, z1))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

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

POL(0) = 0
POL(AND(x₁, x₂)) = 0
POL(ISNAT(x₁)) = 0
POL(ISNATKIND(x₁)) = 0
POL(PLUS(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(TOP(x₁)) = 0
POL(U11'(x₁, x₂, x₃)) = [2]x₃
POL(U12'(x₁, x₂)) = 0
POL(U13'(x₁)) = 0
POL(U21'(x₁, x₂)) = 0
POL(U22'(x₁)) = 0
POL(U31'(x₁, x₂, x₃)) = [2]x₃
POL(U32'(x₁, x₂)) = 0
POL(U33'(x₁)) = 0
POL(U41'(x₁, x₂)) = 0
POL(U51'(x₁, x₂, x₃)) = 0
POL(U61'(x₁)) = 0
POL(U71'(x₁, x₂, x₃)) = 0
POL(X(x₁, x₂)) = 0
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(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(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(mark(x₁)) = 0
POL(ok(x₁)) = [1] + x₁
POL(proper(x₁)) = 0
POL(tt) = 0

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)

Tuples:

U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
U61'(mark(z0)) → c11(U61'(z0))
U61'(ok(z0)) → c12(U61'(z0))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0), z1) → c22(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
X(mark(z0), z1) → c33(X(z0, z1))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
U61'(mark(z0)) → c11(U61'(z0))
U61'(ok(z0)) → c12(U61'(z0))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0), z1) → c22(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
X(mark(z0), z1) → c33(X(z0, z1))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

K tuples:

U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U33', U31', U32', ISNAT, U71', U61', AND, U51', U41', PLUS, U21', ISNATKIND, U11', U13', U22', X, U12', S, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c1

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

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

TOP(mark(z0)) → c1(TOP(proper(z0)))

We considered the (Usable) Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)

And the Tuples:

U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
U61'(mark(z0)) → c11(U61'(z0))
U61'(ok(z0)) → c12(U61'(z0))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0), z1) → c22(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
X(mark(z0), z1) → c33(X(z0, z1))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

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

POL(0) = 0
POL(AND(x₁, x₂)) = 0
POL(ISNAT(x₁)) = 0
POL(ISNATKIND(x₁)) = 0
POL(PLUS(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(TOP(x₁)) = x₁
POL(U11'(x₁, x₂, x₃)) = 0
POL(U12'(x₁, x₂)) = 0
POL(U13'(x₁)) = 0
POL(U21'(x₁, x₂)) = 0
POL(U22'(x₁)) = 0
POL(U31'(x₁, x₂, x₃)) = 0
POL(U32'(x₁, x₂)) = 0
POL(U33'(x₁)) = 0
POL(U41'(x₁, x₂)) = 0
POL(U51'(x₁, x₂, x₃)) = 0
POL(U61'(x₁)) = 0
POL(U71'(x₁, x₂, x₃)) = 0
POL(X(x₁, x₂)) = 0
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(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(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = 0
POL(proper(x₁)) = 0
POL(tt) = 0

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)

Tuples:

U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
U61'(mark(z0)) → c11(U61'(z0))
U61'(ok(z0)) → c12(U61'(z0))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0), z1) → c22(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
X(mark(z0), z1) → c33(X(z0, z1))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
U61'(mark(z0)) → c11(U61'(z0))
U61'(ok(z0)) → c12(U61'(z0))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0), z1) → c22(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
X(mark(z0), z1) → c33(X(z0, z1))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))

K tuples:

U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))
TOP(mark(z0)) → c1(TOP(proper(z0)))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U33', U31', U32', ISNAT, U71', U61', AND, U51', U41', PLUS, U21', ISNATKIND, U11', U13', U22', X, U12', S, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c1

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

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

U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))

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

U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
U61'(mark(z0)) → c11(U61'(z0))
U61'(ok(z0)) → c12(U61'(z0))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0), z1) → c22(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
X(mark(z0), z1) → c33(X(z0, z1))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

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

POL(0) = 0
POL(AND(x₁, x₂)) = x₁
POL(ISNAT(x₁)) = 0
POL(ISNATKIND(x₁)) = x₁
POL(PLUS(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(TOP(x₁)) = 0
POL(U11'(x₁, x₂, x₃)) = x₃
POL(U12'(x₁, x₂)) = x₁
POL(U13'(x₁)) = 0
POL(U21'(x₁, x₂)) = 0
POL(U22'(x₁)) = 0
POL(U31'(x₁, x₂, x₃)) = x₁
POL(U32'(x₁, x₂)) = 0
POL(U33'(x₁)) = 0
POL(U41'(x₁, x₂)) = x₁
POL(U51'(x₁, x₂, x₃)) = x₂ + x₃
POL(U61'(x₁)) = 0
POL(U71'(x₁, x₂, x₃)) = x₁
POL(X(x₁, x₂)) = x₂
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(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(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = [1] + x₁
POL(proper(x₁)) = 0
POL(tt) = 0

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)

Tuples:

U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
U61'(mark(z0)) → c11(U61'(z0))
U61'(ok(z0)) → c12(U61'(z0))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0), z1) → c22(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
X(mark(z0), z1) → c33(X(z0, z1))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U61'(mark(z0)) → c11(U61'(z0))
U61'(ok(z0)) → c12(U61'(z0))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0), z1) → c22(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
X(mark(z0), z1) → c33(X(z0, z1))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))

K tuples:

U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))
TOP(mark(z0)) → c1(TOP(proper(z0)))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U33', U31', U32', ISNAT, U71', U61', AND, U51', U41', PLUS, U21', ISNATKIND, U11', U13', U22', X, U12', S, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c1

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

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

U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))

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

U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
U61'(mark(z0)) → c11(U61'(z0))
U61'(ok(z0)) → c12(U61'(z0))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0), z1) → c22(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
X(mark(z0), z1) → c33(X(z0, z1))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

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

POL(0) = [1]
POL(AND(x₁, x₂)) = 0
POL(ISNAT(x₁)) = 0
POL(ISNATKIND(x₁)) = 0
POL(PLUS(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(TOP(x₁)) = 0
POL(U11'(x₁, x₂, x₃)) = x₁
POL(U12'(x₁, x₂)) = 0
POL(U13'(x₁)) = 0
POL(U21'(x₁, x₂)) = 0
POL(U22'(x₁)) = 0
POL(U31'(x₁, x₂, x₃)) = x₃
POL(U32'(x₁, x₂)) = 0
POL(U33'(x₁)) = 0
POL(U41'(x₁, x₂)) = x₁
POL(U51'(x₁, x₂, x₃)) = 0
POL(U61'(x₁)) = 0
POL(U71'(x₁, x₂, x₃)) = 0
POL(X(x₁, x₂)) = x₂
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(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(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = x₁
POL(proper(x₁)) = [1]
POL(tt) = 0

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)

Tuples:

U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
U61'(mark(z0)) → c11(U61'(z0))
U61'(ok(z0)) → c12(U61'(z0))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0), z1) → c22(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
X(mark(z0), z1) → c33(X(z0, z1))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U61'(mark(z0)) → c11(U61'(z0))
U61'(ok(z0)) → c12(U61'(z0))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0), z1) → c22(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
X(mark(z0), z1) → c33(X(z0, z1))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))

K tuples:

U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))
TOP(mark(z0)) → c1(TOP(proper(z0)))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U33', U31', U32', ISNAT, U71', U61', AND, U51', U41', PLUS, U21', ISNATKIND, U11', U13', U22', X, U12', S, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c1

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

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

ISNAT(ok(z0)) → c8(ISNAT(z0))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
X(mark(z0), z1) → c33(X(z0, z1))

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

U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
U61'(mark(z0)) → c11(U61'(z0))
U61'(ok(z0)) → c12(U61'(z0))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0), z1) → c22(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
X(mark(z0), z1) → c33(X(z0, z1))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

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

POL(0) = 0
POL(AND(x₁, x₂)) = [2]x₂
POL(ISNAT(x₁)) = x₁
POL(ISNATKIND(x₁)) = [2]x₁
POL(PLUS(x₁, x₂)) = x₁
POL(S(x₁)) = 0
POL(TOP(x₁)) = 0
POL(U11'(x₁, x₂, x₃)) = x₂
POL(U12'(x₁, x₂)) = [2]x₂
POL(U13'(x₁)) = 0
POL(U21'(x₁, x₂)) = 0
POL(U22'(x₁)) = 0
POL(U31'(x₁, x₂, x₃)) = [2]x₂ + [3]x₃
POL(U32'(x₁, x₂)) = 0
POL(U33'(x₁)) = 0
POL(U41'(x₁, x₂)) = 0
POL(U51'(x₁, x₂, x₃)) = x₁ + [3]x₂
POL(U61'(x₁)) = 0
POL(U71'(x₁, x₂, x₃)) = [2]x₁ + [2]x₂
POL(X(x₁, x₂)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(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(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = [2] + x₁
POL(proper(x₁)) = 0
POL(tt) = 0

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)

Tuples:

U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
U61'(mark(z0)) → c11(U61'(z0))
U61'(ok(z0)) → c12(U61'(z0))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0), z1) → c22(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
X(mark(z0), z1) → c33(X(z0, z1))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
U61'(mark(z0)) → c11(U61'(z0))
U61'(ok(z0)) → c12(U61'(z0))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
U21'(mark(z0), z1) → c22(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))

K tuples:

U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))
TOP(mark(z0)) → c1(TOP(proper(z0)))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
X(mark(z0), z1) → c33(X(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U33', U31', U32', ISNAT, U71', U61', AND, U51', U41', PLUS, U21', ISNATKIND, U11', U13', U22', X, U12', S, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c1

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

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

U32'(mark(z0), z1) → c6(U32'(z0, z1))

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

U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
U61'(mark(z0)) → c11(U61'(z0))
U61'(ok(z0)) → c12(U61'(z0))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0), z1) → c22(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
X(mark(z0), z1) → c33(X(z0, z1))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

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

POL(0) = [1]
POL(AND(x₁, x₂)) = 0
POL(ISNAT(x₁)) = 0
POL(ISNATKIND(x₁)) = 0
POL(PLUS(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(TOP(x₁)) = 0
POL(U11'(x₁, x₂, x₃)) = 0
POL(U12'(x₁, x₂)) = 0
POL(U13'(x₁)) = 0
POL(U21'(x₁, x₂)) = 0
POL(U22'(x₁)) = 0
POL(U31'(x₁, x₂, x₃)) = 0
POL(U32'(x₁, x₂)) = x₁
POL(U33'(x₁)) = 0
POL(U41'(x₁, x₂)) = 0
POL(U51'(x₁, x₂, x₃)) = 0
POL(U61'(x₁)) = 0
POL(U71'(x₁, x₂, x₃)) = 0
POL(X(x₁, x₂)) = 0
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(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(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = x₁
POL(proper(x₁)) = [1] + x₁
POL(tt) = 0

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)

Tuples:

U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
U61'(mark(z0)) → c11(U61'(z0))
U61'(ok(z0)) → c12(U61'(z0))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0), z1) → c22(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
X(mark(z0), z1) → c33(X(z0, z1))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
U61'(mark(z0)) → c11(U61'(z0))
U61'(ok(z0)) → c12(U61'(z0))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
U21'(mark(z0), z1) → c22(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))

K tuples:

U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))
TOP(mark(z0)) → c1(TOP(proper(z0)))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
X(mark(z0), z1) → c33(X(z0, z1))
U32'(mark(z0), z1) → c6(U32'(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U33', U31', U32', ISNAT, U71', U61', AND, U51', U41', PLUS, U21', ISNATKIND, U11', U13', U22', X, U12', S, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c1

(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.

PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
S(mark(z0)) → c39(S(z0))

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

U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
U61'(mark(z0)) → c11(U61'(z0))
U61'(ok(z0)) → c12(U61'(z0))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0), z1) → c22(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
X(mark(z0), z1) → c33(X(z0, z1))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

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

POL(0) = 0
POL(AND(x₁, x₂)) = 0
POL(ISNAT(x₁)) = 0
POL(ISNATKIND(x₁)) = 0
POL(PLUS(x₁, x₂)) = x₁ + x₂
POL(S(x₁)) = x₁
POL(TOP(x₁)) = 0
POL(U11'(x₁, x₂, x₃)) = 0
POL(U12'(x₁, x₂)) = 0
POL(U13'(x₁)) = 0
POL(U21'(x₁, x₂)) = 0
POL(U22'(x₁)) = 0
POL(U31'(x₁, x₂, x₃)) = 0
POL(U32'(x₁, x₂)) = 0
POL(U33'(x₁)) = 0
POL(U41'(x₁, x₂)) = 0
POL(U51'(x₁, x₂, x₃)) = 0
POL(U61'(x₁)) = 0
POL(U71'(x₁, x₂, x₃)) = 0
POL(X(x₁, x₂)) = 0
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(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(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 0
POL(tt) = 0

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)

Tuples:

U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
U61'(mark(z0)) → c11(U61'(z0))
U61'(ok(z0)) → c12(U61'(z0))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0), z1) → c22(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
X(mark(z0), z1) → c33(X(z0, z1))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
U61'(mark(z0)) → c11(U61'(z0))
U61'(ok(z0)) → c12(U61'(z0))
U21'(mark(z0), z1) → c22(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
S(ok(z0)) → c38(S(z0))

K tuples:

U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))
TOP(mark(z0)) → c1(TOP(proper(z0)))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
X(mark(z0), z1) → c33(X(z0, z1))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
S(mark(z0)) → c39(S(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U33', U31', U32', ISNAT, U71', U61', AND, U51', U41', PLUS, U21', ISNATKIND, U11', U13', U22', X, U12', S, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c1

(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.

U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))

We considered the (Usable) Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)

And the Tuples:

U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
U61'(mark(z0)) → c11(U61'(z0))
U61'(ok(z0)) → c12(U61'(z0))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0), z1) → c22(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
X(mark(z0), z1) → c33(X(z0, z1))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

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

POL(0) = 0
POL(AND(x₁, x₂)) = 0
POL(ISNAT(x₁)) = x₁
POL(ISNATKIND(x₁)) = 0
POL(PLUS(x₁, x₂)) = x₂
POL(S(x₁)) = 0
POL(TOP(x₁)) = x₁
POL(U11'(x₁, x₂, x₃)) = 0
POL(U12'(x₁, x₂)) = 0
POL(U13'(x₁)) = x₁
POL(U21'(x₁, x₂)) = 0
POL(U22'(x₁)) = 0
POL(U31'(x₁, x₂, x₃)) = 0
POL(U32'(x₁, x₂)) = 0
POL(U33'(x₁)) = x₁
POL(U41'(x₁, x₂)) = 0
POL(U51'(x₁, x₂, x₃)) = 0
POL(U61'(x₁)) = 0
POL(U71'(x₁, x₂, x₃)) = 0
POL(X(x₁, x₂)) = 0
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(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(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = [1] + x₁
POL(proper(x₁)) = [1] + x₁
POL(tt) = 0

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)

Tuples:

U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
U61'(mark(z0)) → c11(U61'(z0))
U61'(ok(z0)) → c12(U61'(z0))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0), z1) → c22(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
X(mark(z0), z1) → c33(X(z0, z1))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
U61'(mark(z0)) → c11(U61'(z0))
U61'(ok(z0)) → c12(U61'(z0))
U21'(mark(z0), z1) → c22(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
S(ok(z0)) → c38(S(z0))

K tuples:

U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))
TOP(mark(z0)) → c1(TOP(proper(z0)))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
X(mark(z0), z1) → c33(X(z0, z1))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
S(mark(z0)) → c39(S(z0))
U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U33', U31', U32', ISNAT, U71', U61', AND, U51', U41', PLUS, U21', ISNATKIND, U11', U13', U22', X, U12', S, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c1

(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.

U21'(mark(z0), z1) → c22(U21'(z0, z1))

We considered the (Usable) Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)

And the Tuples:

U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
U61'(mark(z0)) → c11(U61'(z0))
U61'(ok(z0)) → c12(U61'(z0))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0), z1) → c22(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
X(mark(z0), z1) → c33(X(z0, z1))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

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

POL(0) = 0
POL(AND(x₁, x₂)) = 0
POL(ISNAT(x₁)) = 0
POL(ISNATKIND(x₁)) = 0
POL(PLUS(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(TOP(x₁)) = x₁
POL(U11'(x₁, x₂, x₃)) = 0
POL(U12'(x₁, x₂)) = 0
POL(U13'(x₁)) = 0
POL(U21'(x₁, x₂)) = x₁
POL(U22'(x₁)) = 0
POL(U31'(x₁, x₂, x₃)) = x₃
POL(U32'(x₁, x₂)) = 0
POL(U33'(x₁)) = 0
POL(U41'(x₁, x₂)) = 0
POL(U51'(x₁, x₂, x₃)) = 0
POL(U61'(x₁)) = 0
POL(U71'(x₁, x₂, x₃)) = x₂
POL(X(x₁, x₂)) = 0
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(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(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = x₁
POL(proper(x₁)) = [1]
POL(tt) = 0

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)

Tuples:

U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
U61'(mark(z0)) → c11(U61'(z0))
U61'(ok(z0)) → c12(U61'(z0))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0), z1) → c22(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
X(mark(z0), z1) → c33(X(z0, z1))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
U61'(mark(z0)) → c11(U61'(z0))
U61'(ok(z0)) → c12(U61'(z0))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
S(ok(z0)) → c38(S(z0))

K tuples:

U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))
TOP(mark(z0)) → c1(TOP(proper(z0)))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
X(mark(z0), z1) → c33(X(z0, z1))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
S(mark(z0)) → c39(S(z0))
U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))
U21'(mark(z0), z1) → c22(U21'(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U33', U31', U32', ISNAT, U71', U61', AND, U51', U41', PLUS, U21', ISNATKIND, U11', U13', U22', X, U12', S, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c1

(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.

U22'(mark(z0)) → c31(U22'(z0))

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

U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
U61'(mark(z0)) → c11(U61'(z0))
U61'(ok(z0)) → c12(U61'(z0))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0), z1) → c22(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
X(mark(z0), z1) → c33(X(z0, z1))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

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

POL(0) = 0
POL(AND(x₁, x₂)) = 0
POL(ISNAT(x₁)) = 0
POL(ISNATKIND(x₁)) = 0
POL(PLUS(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(TOP(x₁)) = 0
POL(U11'(x₁, x₂, x₃)) = 0
POL(U12'(x₁, x₂)) = 0
POL(U13'(x₁)) = x₁
POL(U21'(x₁, x₂)) = 0
POL(U22'(x₁)) = x₁
POL(U31'(x₁, x₂, x₃)) = x₂
POL(U32'(x₁, x₂)) = 0
POL(U33'(x₁)) = 0
POL(U41'(x₁, x₂)) = 0
POL(U51'(x₁, x₂, x₃)) = x₃
POL(U61'(x₁)) = 0
POL(U71'(x₁, x₂, x₃)) = 0
POL(X(x₁, x₂)) = 0
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(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(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 0
POL(tt) = 0

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)

Tuples:

U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
U61'(mark(z0)) → c11(U61'(z0))
U61'(ok(z0)) → c12(U61'(z0))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0), z1) → c22(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
X(mark(z0), z1) → c33(X(z0, z1))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
U61'(mark(z0)) → c11(U61'(z0))
U61'(ok(z0)) → c12(U61'(z0))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
U22'(ok(z0)) → c32(U22'(z0))
S(ok(z0)) → c38(S(z0))

K tuples:

U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))
TOP(mark(z0)) → c1(TOP(proper(z0)))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
X(mark(z0), z1) → c33(X(z0, z1))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
S(mark(z0)) → c39(S(z0))
U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))
U21'(mark(z0), z1) → c22(U21'(z0, z1))
U22'(mark(z0)) → c31(U22'(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U33', U31', U32', ISNAT, U71', U61', AND, U51', U41', PLUS, U21', ISNATKIND, U11', U13', U22', X, U12', S, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c1

(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.

U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))

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

U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
U61'(mark(z0)) → c11(U61'(z0))
U61'(ok(z0)) → c12(U61'(z0))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0), z1) → c22(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
X(mark(z0), z1) → c33(X(z0, z1))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

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

POL(0) = [1]
POL(AND(x₁, x₂)) = 0
POL(ISNAT(x₁)) = 0
POL(ISNATKIND(x₁)) = [2]x₁
POL(PLUS(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(TOP(x₁)) = 0
POL(U11'(x₁, x₂, x₃)) = [2]x₂
POL(U12'(x₁, x₂)) = 0
POL(U13'(x₁)) = 0
POL(U21'(x₁, x₂)) = 0
POL(U22'(x₁)) = 0
POL(U31'(x₁, x₂, x₃)) = x₂
POL(U32'(x₁, x₂)) = x₂
POL(U33'(x₁)) = 0
POL(U41'(x₁, x₂)) = 0
POL(U51'(x₁, x₂, x₃)) = [2]x₃
POL(U61'(x₁)) = 0
POL(U71'(x₁, x₂, x₃)) = 0
POL(X(x₁, x₂)) = 0
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(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(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(mark(x₁)) = 0
POL(ok(x₁)) = [1] + x₁
POL(proper(x₁)) = [3]x₁
POL(tt) = [3]

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)

Tuples:

U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
U61'(mark(z0)) → c11(U61'(z0))
U61'(ok(z0)) → c12(U61'(z0))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0), z1) → c22(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
X(mark(z0), z1) → c33(X(z0, z1))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

U61'(mark(z0)) → c11(U61'(z0))
U61'(ok(z0)) → c12(U61'(z0))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
U22'(ok(z0)) → c32(U22'(z0))
S(ok(z0)) → c38(S(z0))

K tuples:

U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))
TOP(mark(z0)) → c1(TOP(proper(z0)))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
X(mark(z0), z1) → c33(X(z0, z1))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
S(mark(z0)) → c39(S(z0))
U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))
U21'(mark(z0), z1) → c22(U21'(z0, z1))
U22'(mark(z0)) → c31(U22'(z0))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U33', U31', U32', ISNAT, U71', U61', AND, U51', U41', PLUS, U21', ISNATKIND, U11', U13', U22', X, U12', S, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c1

(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.

U61'(mark(z0)) → c11(U61'(z0))

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

U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
U61'(mark(z0)) → c11(U61'(z0))
U61'(ok(z0)) → c12(U61'(z0))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0), z1) → c22(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
X(mark(z0), z1) → c33(X(z0, z1))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

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

POL(0) = 0
POL(AND(x₁, x₂)) = 0
POL(ISNAT(x₁)) = 0
POL(ISNATKIND(x₁)) = 0
POL(PLUS(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(TOP(x₁)) = 0
POL(U11'(x₁, x₂, x₃)) = 0
POL(U12'(x₁, x₂)) = 0
POL(U13'(x₁)) = 0
POL(U21'(x₁, x₂)) = 0
POL(U22'(x₁)) = 0
POL(U31'(x₁, x₂, x₃)) = 0
POL(U32'(x₁, x₂)) = 0
POL(U33'(x₁)) = 0
POL(U41'(x₁, x₂)) = 0
POL(U51'(x₁, x₂, x₃)) = 0
POL(U61'(x₁)) = x₁
POL(U71'(x₁, x₂, x₃)) = x₂
POL(X(x₁, x₂)) = 0
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(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(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 0
POL(tt) = 0

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)

Tuples:

U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
U61'(mark(z0)) → c11(U61'(z0))
U61'(ok(z0)) → c12(U61'(z0))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0), z1) → c22(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
X(mark(z0), z1) → c33(X(z0, z1))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

U61'(ok(z0)) → c12(U61'(z0))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
U22'(ok(z0)) → c32(U22'(z0))
S(ok(z0)) → c38(S(z0))

K tuples:

U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))
TOP(mark(z0)) → c1(TOP(proper(z0)))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
X(mark(z0), z1) → c33(X(z0, z1))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
S(mark(z0)) → c39(S(z0))
U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))
U21'(mark(z0), z1) → c22(U21'(z0, z1))
U22'(mark(z0)) → c31(U22'(z0))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
U61'(mark(z0)) → c11(U61'(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U33', U31', U32', ISNAT, U71', U61', AND, U51', U41', PLUS, U21', ISNATKIND, U11', U13', U22', X, U12', S, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c1

(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'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
S(ok(z0)) → c38(S(z0))

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

U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
U61'(mark(z0)) → c11(U61'(z0))
U61'(ok(z0)) → c12(U61'(z0))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0), z1) → c22(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
X(mark(z0), z1) → c33(X(z0, z1))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

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

POL(0) = 0
POL(AND(x₁, x₂)) = 0
POL(ISNAT(x₁)) = 0
POL(ISNATKIND(x₁)) = 0
POL(PLUS(x₁, x₂)) = 0
POL(S(x₁)) = x₁
POL(TOP(x₁)) = 0
POL(U11'(x₁, x₂, x₃)) = 0
POL(U12'(x₁, x₂)) = 0
POL(U13'(x₁)) = 0
POL(U21'(x₁, x₂)) = x₁
POL(U22'(x₁)) = 0
POL(U31'(x₁, x₂, x₃)) = 0
POL(U32'(x₁, x₂)) = 0
POL(U33'(x₁)) = 0
POL(U41'(x₁, x₂)) = x₂
POL(U51'(x₁, x₂, x₃)) = 0
POL(U61'(x₁)) = 0
POL(U71'(x₁, x₂, x₃)) = 0
POL(X(x₁, x₂)) = 0
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(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(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(ok(x₁)) = [1] + x₁
POL(proper(x₁)) = 0
POL(tt) = 0

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)

Tuples:

U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
U61'(mark(z0)) → c11(U61'(z0))
U61'(ok(z0)) → c12(U61'(z0))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0), z1) → c22(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
X(mark(z0), z1) → c33(X(z0, z1))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

U61'(ok(z0)) → c12(U61'(z0))
U22'(ok(z0)) → c32(U22'(z0))

K tuples:

U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))
TOP(mark(z0)) → c1(TOP(proper(z0)))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
X(mark(z0), z1) → c33(X(z0, z1))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
S(mark(z0)) → c39(S(z0))
U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))
U21'(mark(z0), z1) → c22(U21'(z0, z1))
U22'(mark(z0)) → c31(U22'(z0))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
U61'(mark(z0)) → c11(U61'(z0))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
S(ok(z0)) → c38(S(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U33', U31', U32', ISNAT, U71', U61', AND, U51', U41', PLUS, U21', ISNATKIND, U11', U13', U22', X, U12', S, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c1

(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.

U61'(ok(z0)) → c12(U61'(z0))

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

U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
U61'(mark(z0)) → c11(U61'(z0))
U61'(ok(z0)) → c12(U61'(z0))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0), z1) → c22(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
X(mark(z0), z1) → c33(X(z0, z1))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

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

POL(0) = 0
POL(AND(x₁, x₂)) = 0
POL(ISNAT(x₁)) = 0
POL(ISNATKIND(x₁)) = 0
POL(PLUS(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(TOP(x₁)) = 0
POL(U11'(x₁, x₂, x₃)) = 0
POL(U12'(x₁, x₂)) = 0
POL(U13'(x₁)) = 0
POL(U21'(x₁, x₂)) = 0
POL(U22'(x₁)) = 0
POL(U31'(x₁, x₂, x₃)) = x₂
POL(U32'(x₁, x₂)) = 0
POL(U33'(x₁)) = 0
POL(U41'(x₁, x₂)) = x₂
POL(U51'(x₁, x₂, x₃)) = 0
POL(U61'(x₁)) = x₁
POL(U71'(x₁, x₂, x₃)) = 0
POL(X(x₁, x₂)) = 0
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(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(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(ok(x₁)) = [1] + x₁
POL(proper(x₁)) = 0
POL(tt) = 0

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)

Tuples:

U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
U61'(mark(z0)) → c11(U61'(z0))
U61'(ok(z0)) → c12(U61'(z0))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0), z1) → c22(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
X(mark(z0), z1) → c33(X(z0, z1))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:

U22'(ok(z0)) → c32(U22'(z0))

K tuples:

U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))
TOP(mark(z0)) → c1(TOP(proper(z0)))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
X(mark(z0), z1) → c33(X(z0, z1))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
S(mark(z0)) → c39(S(z0))
U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))
U21'(mark(z0), z1) → c22(U21'(z0, z1))
U22'(mark(z0)) → c31(U22'(z0))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
U61'(mark(z0)) → c11(U61'(z0))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
S(ok(z0)) → c38(S(z0))
U61'(ok(z0)) → c12(U61'(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U33', U31', U32', ISNAT, U71', U61', AND, U51', U41', PLUS, U21', ISNATKIND, U11', U13', U22', X, U12', S, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c1

(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.

U22'(ok(z0)) → c32(U22'(z0))

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

U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
U61'(mark(z0)) → c11(U61'(z0))
U61'(ok(z0)) → c12(U61'(z0))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0), z1) → c22(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
X(mark(z0), z1) → c33(X(z0, z1))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

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

POL(0) = 0
POL(AND(x₁, x₂)) = 0
POL(ISNAT(x₁)) = 0
POL(ISNATKIND(x₁)) = 0
POL(PLUS(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(TOP(x₁)) = 0
POL(U11'(x₁, x₂, x₃)) = 0
POL(U12'(x₁, x₂)) = 0
POL(U13'(x₁)) = 0
POL(U21'(x₁, x₂)) = 0
POL(U22'(x₁)) = x₁
POL(U31'(x₁, x₂, x₃)) = 0
POL(U32'(x₁, x₂)) = 0
POL(U33'(x₁)) = 0
POL(U41'(x₁, x₂)) = 0
POL(U51'(x₁, x₂, x₃)) = 0
POL(U61'(x₁)) = 0
POL(U71'(x₁, x₂, x₃)) = 0
POL(X(x₁, x₂)) = 0
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(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(c5(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c8(x₁)) = x₁
POL(c9(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(ok(x₁)) = [1] + x₁
POL(proper(x₁)) = 0
POL(tt) = 0

(44) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)

Tuples:

U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
U61'(mark(z0)) → c11(U61'(z0))
U61'(ok(z0)) → c12(U61'(z0))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
U21'(mark(z0), z1) → c22(U21'(z0, z1))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))
U22'(mark(z0)) → c31(U22'(z0))
U22'(ok(z0)) → c32(U22'(z0))
X(mark(z0), z1) → c33(X(z0, z1))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))
S(ok(z0)) → c38(S(z0))
S(mark(z0)) → c39(S(z0))
TOP(mark(z0)) → c1(TOP(proper(z0)))

S tuples:none
K tuples:

U11'(ok(z0), ok(z1), ok(z2)) → c28(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c4(U31'(z0, z1, z2))
TOP(mark(z0)) → c1(TOP(proper(z0)))
U31'(mark(z0), z1, z2) → c5(U31'(z0, z1, z2))
U71'(ok(z0), ok(z1), ok(z2)) → c9(U71'(z0, z1, z2))
U71'(mark(z0), z1, z2) → c10(U71'(z0, z1, z2))
AND(ok(z0), ok(z1)) → c13(AND(z0, z1))
AND(mark(z0), z1) → c14(AND(z0, z1))
U51'(ok(z0), ok(z1), ok(z2)) → c16(U51'(z0, z1, z2))
U41'(mark(z0), z1) → c17(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c18(U41'(z0, z1))
ISNATKIND(ok(z0)) → c26(ISNATKIND(z0))
X(ok(z0), ok(z1)) → c34(X(z0, z1))
X(z0, mark(z1)) → c35(X(z0, z1))
U12'(ok(z0), ok(z1)) → c36(U12'(z0, z1))
U12'(mark(z0), z1) → c37(U12'(z0, z1))
U11'(mark(z0), z1, z2) → c27(U11'(z0, z1, z2))
ISNAT(ok(z0)) → c8(ISNAT(z0))
U51'(mark(z0), z1, z2) → c15(U51'(z0, z1, z2))
PLUS(ok(z0), ok(z1)) → c19(PLUS(z0, z1))
PLUS(mark(z0), z1) → c21(PLUS(z0, z1))
X(mark(z0), z1) → c33(X(z0, z1))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
PLUS(z0, mark(z1)) → c20(PLUS(z0, z1))
S(mark(z0)) → c39(S(z0))
U33'(ok(z0)) → c2(U33'(z0))
U33'(mark(z0)) → c3(U33'(z0))
U13'(ok(z0)) → c29(U13'(z0))
U13'(mark(z0)) → c30(U13'(z0))
U21'(mark(z0), z1) → c22(U21'(z0, z1))
U22'(mark(z0)) → c31(U22'(z0))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
U61'(mark(z0)) → c11(U61'(z0))
U21'(ok(z0), ok(z1)) → c23(U21'(z0, z1))
S(ok(z0)) → c38(S(z0))
U61'(ok(z0)) → c12(U61'(z0))
U22'(ok(z0)) → c32(U22'(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U33', U31', U32', ISNAT, U71', U61', AND, U51', U41', PLUS, U21', ISNATKIND, U11', U13', U22', X, U12', S, TOP

Compound Symbols:

c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c1

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

(46) BOUNDS(1, 1)