Runtime Complexity (innermost) proof of /tmp/tmpP5yXHZ/MYNAT_complete_noand

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

0 CpxTRS

↳1 NestedDefinedSymbolProof (BOTH BOUNDS(ID, ID), 41 ms)

↳2 CpxTRS

    ↳3 CpxTrsMatchBoundsTAProof (⇔, 2424 ms)

    ↳4 BOUNDS(1, n^1)

    ↳5 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳6 CdtProblem

        ↳7 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)

        ↳8 CdtProblem

        ↳9 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

        ↳10 CdtProblem

        ↳11 CdtUsableRulesProof (⇔, 14 ms)

        ↳12 CdtProblem

        ↳13 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 374 ms)

        ↳14 CdtProblem

        ↳15 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 105 ms)

        ↳16 CdtProblem

        ↳17 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 98 ms)

        ↳18 CdtProblem

        ↳19 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 114 ms)

        ↳20 CdtProblem

        ↳21 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 114 ms)

        ↳22 CdtProblem

        ↳23 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 85 ms)

        ↳24 CdtProblem

        ↳25 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 75 ms)

        ↳26 CdtProblem

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

        ↳28 CdtProblem

        ↳29 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 70 ms)

        ↳30 CdtProblem

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

        ↳32 CdtProblem

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

        ↳34 CdtProblem

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

        ↳36 CdtProblem

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

        ↳38 CdtProblem

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

        ↳40 CdtProblem

        ↳41 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 74 ms)

        ↳42 CdtProblem

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

        ↳44 CdtProblem

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

        ↳46 CdtProblem

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

        ↳48 CdtProblem

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

        ↳50 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(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U13(X1, X2, X3)) → U13(active(X1), X2, X3)
active(U14(X1, X2, X3)) → U14(active(X1), X2, X3)
active(U15(X1, X2)) → U15(active(X1), X2)
active(U16(X)) → U16(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2, X3)) → U32(active(X1), X2, X3)
active(U33(X1, X2, X3)) → U33(active(X1), X2, X3)
active(U34(X1, X2, X3)) → U34(active(X1), X2, X3)
active(U35(X1, X2)) → U35(active(X1), X2)
active(U36(X)) → U36(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X)) → U51(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U13(mark(X1), X2, X3) → mark(U13(X1, X2, X3))
U14(mark(X1), X2, X3) → mark(U14(X1, X2, X3))
U15(mark(X1), X2) → mark(U15(X1, X2))
U16(mark(X)) → mark(U16(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2, X3) → mark(U32(X1, X2, X3))
U33(mark(X1), X2, X3) → mark(U33(X1, X2, X3))
U34(mark(X1), X2, X3) → mark(U34(X1, X2, X3))
U35(mark(X1), X2) → mark(U35(X1, X2))
U36(mark(X)) → mark(U36(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X)) → mark(U51(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
s(mark(X)) → mark(s(X))
U91(mark(X1), X2) → mark(U91(X1, X2))
U92(mark(X)) → mark(U92(X))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNat(X)) → isNat(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U13(X1, X2, X3)) → U13(proper(X1), proper(X2), proper(X3))
proper(U14(X1, X2, X3)) → U14(proper(X1), proper(X2), proper(X3))
proper(U15(X1, X2)) → U15(proper(X1), proper(X2))
proper(U16(X)) → U16(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2, X3)) → U32(proper(X1), proper(X2), proper(X3))
proper(U33(X1, X2, X3)) → U33(proper(X1), proper(X2), proper(X3))
proper(U34(X1, X2, X3)) → U34(proper(X1), proper(X2), proper(X3))
proper(U35(X1, X2)) → U35(proper(X1), proper(X2))
proper(U36(X)) → U36(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X)) → U51(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(proper(X))
proper(0) → ok(0)
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
isNat(ok(X)) → ok(isNat(X))
U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U13(ok(X1), ok(X2), ok(X3)) → ok(U13(X1, X2, X3))
U14(ok(X1), ok(X2), ok(X3)) → ok(U14(X1, X2, X3))
U15(ok(X1), ok(X2)) → ok(U15(X1, X2))
U16(ok(X)) → ok(U16(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2), ok(X3)) → ok(U32(X1, X2, X3))
U33(ok(X1), ok(X2), ok(X3)) → ok(U33(X1, X2, X3))
U34(ok(X1), ok(X2), ok(X3)) → ok(U34(X1, X2, X3))
U35(ok(X1), ok(X2)) → ok(U35(X1, X2))
U36(ok(X)) → ok(U36(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
U51(ok(X)) → ok(U51(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
s(ok(X)) → ok(s(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))
U92(ok(X)) → ok(U92(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(U101(tt, M, N)) → mark(U102(isNatKind(M), M, N))
active(U102(tt, M, N)) → mark(U103(isNat(N), M, N))
active(U103(tt, M, N)) → mark(U104(isNatKind(N), M, N))
active(U104(tt, M, N)) → mark(plus(x(N, M), N))
active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNatKind(V1), V1, V2))
active(U32(tt, V1, V2)) → mark(U33(isNatKind(V2), V1, V2))
active(U33(tt, V1, V2)) → mark(U34(isNatKind(V2), V1, V2))
active(U34(tt, V1, V2)) → mark(U35(isNat(V1), V2))
active(U35(tt, V2)) → mark(U36(isNat(V2)))
active(U36(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatKind(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatKind(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, N)) → mark(U72(isNatKind(N), N))
active(U72(tt, N)) → mark(N)
active(U81(tt, M, N)) → mark(U82(isNatKind(M), M, N))
active(U82(tt, M, N)) → mark(U83(isNat(N), M, N))
active(U83(tt, M, N)) → mark(U84(isNatKind(N), M, N))
active(U84(tt, M, N)) → mark(s(plus(N, M)))
active(U91(tt, N)) → mark(U92(isNatKind(N)))
active(U92(tt)) → mark(0)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(isNatKind(V1), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U41(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U51(isNatKind(V1)))
active(isNatKind(x(V1, V2))) → mark(U61(isNatKind(V1), V2))
active(plus(N, 0)) → mark(U71(isNat(N), N))
active(plus(N, s(M))) → mark(U81(isNat(M), M, N))
active(x(N, 0)) → mark(U91(isNat(N), N))
active(x(N, s(M))) → mark(U101(isNat(M), M, N))
active(U101(X1, X2, X3)) → U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) → U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) → U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) → U104(active(X1), X2, X3)
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U13(X1, X2, X3)) → U13(active(X1), X2, X3)
active(U14(X1, X2, X3)) → U14(active(X1), X2, X3)
active(U15(X1, X2)) → U15(active(X1), X2)
active(U16(X)) → U16(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2, X3)) → U32(active(X1), X2, X3)
active(U33(X1, X2, X3)) → U33(active(X1), X2, X3)
active(U34(X1, X2, X3)) → U34(active(X1), X2, X3)
active(U35(X1, X2)) → U35(active(X1), X2)
active(U36(X)) → U36(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X)) → U51(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(U91(X1, X2)) → U91(active(X1), X2)
active(U92(X)) → U92(active(X))
proper(U101(X1, X2, X3)) → U101(proper(X1), proper(X2), proper(X3))
proper(U102(X1, X2, X3)) → U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U103(X1, X2, X3)) → U103(proper(X1), proper(X2), proper(X3))
proper(isNat(X)) → isNat(proper(X))
proper(U104(X1, X2, X3)) → U104(proper(X1), proper(X2), proper(X3))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(U13(X1, X2, X3)) → U13(proper(X1), proper(X2), proper(X3))
proper(U14(X1, X2, X3)) → U14(proper(X1), proper(X2), proper(X3))
proper(U15(X1, X2)) → U15(proper(X1), proper(X2))
proper(U16(X)) → U16(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2, X3)) → U32(proper(X1), proper(X2), proper(X3))
proper(U33(X1, X2, X3)) → U33(proper(X1), proper(X2), proper(X3))
proper(U34(X1, X2, X3)) → U34(proper(X1), proper(X2), proper(X3))
proper(U35(X1, X2)) → U35(proper(X1), proper(X2))
proper(U36(X)) → U36(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(U51(X)) → U51(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(U91(X1, X2)) → U91(proper(X1), proper(X2))
proper(U92(X)) → U92(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:

U104(ok(X1), ok(X2), ok(X3)) → ok(U104(X1, X2, X3))
top(ok(X)) → top(active(X))
U35(ok(X1), ok(X2)) → ok(U35(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U91(mark(X1), X2) → mark(U91(X1, X2))
U104(mark(X1), X2, X3) → mark(U104(X1, X2, X3))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U36(mark(X)) → mark(U36(X))
isNat(ok(X)) → ok(isNat(X))
U101(ok(X1), ok(X2), ok(X3)) → ok(U101(X1, X2, X3))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U92(mark(X)) → mark(U92(X))
U62(mark(X)) → mark(U62(X))
U35(mark(X1), X2) → mark(U35(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U42(mark(X)) → mark(U42(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U103(mark(X1), X2, X3) → mark(U103(X1, X2, X3))
proper(tt) → ok(tt)
isNatKind(ok(X)) → ok(isNatKind(X))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U51(mark(X)) → mark(U51(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U15(ok(X1), ok(X2)) → ok(U15(X1, X2))
U61(mark(X1), X2) → mark(U61(X1, X2))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
U32(ok(X1), ok(X2), ok(X3)) → ok(U32(X1, X2, X3))
U13(mark(X1), X2, X3) → mark(U13(X1, X2, X3))
U16(mark(X)) → mark(U16(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U36(ok(X)) → ok(U36(X))
s(ok(X)) → ok(s(X))
U23(ok(X)) → ok(U23(X))
U101(mark(X1), X2, X3) → mark(U101(X1, X2, X3))
U34(ok(X1), ok(X2), ok(X3)) → ok(U34(X1, X2, X3))
plus(mark(X1), X2) → mark(plus(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
proper(0) → ok(0)
U13(ok(X1), ok(X2), ok(X3)) → ok(U13(X1, X2, X3))
U34(mark(X1), X2, X3) → mark(U34(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U22(mark(X1), X2) → mark(U22(X1, X2))
U102(mark(X1), X2, X3) → mark(U102(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U92(ok(X)) → ok(U92(X))
U62(ok(X)) → ok(U62(X))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U32(mark(X1), X2, X3) → mark(U32(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) → ok(U102(X1, X2, X3))
U103(ok(X1), ok(X2), ok(X3)) → ok(U103(X1, X2, X3))
U42(ok(X)) → ok(U42(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
x(mark(X1), X2) → mark(x(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U23(mark(X)) → mark(U23(X))
U14(ok(X1), ok(X2), ok(X3)) → ok(U14(X1, X2, X3))
U15(mark(X1), X2) → mark(U15(X1, X2))
U16(ok(X)) → ok(U16(X))
U33(mark(X1), X2, X3) → mark(U33(X1, X2, X3))
U33(ok(X1), ok(X2), ok(X3)) → ok(U33(X1, X2, X3))
U51(ok(X)) → ok(U51(X))
s(mark(X)) → mark(s(X))
x(X1, mark(X2)) → mark(x(X1, X2))
U14(mark(X1), X2, X3) → mark(U14(X1, X2, X3))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
top(mark(X)) → top(proper(X))
U91(ok(X1), ok(X2)) → ok(U91(X1, X2))

Rewrite Strategy: INNERMOST

(3) CpxTrsMatchBoundsTAProof (EQUIVALENT transformation)

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

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by:
final states : [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39]
transitions:
ok0(0) → 0
active0(0) → 0
mark0(0) → 0
tt0() → 0
00() → 0
U1040(0, 0, 0) → 1
top0(0) → 2
U350(0, 0) → 3
U810(0, 0, 0) → 4
U310(0, 0, 0) → 5
U910(0, 0) → 6
U360(0) → 7
isNat0(0) → 8
U1010(0, 0, 0) → 9
U610(0, 0) → 10
U840(0, 0, 0) → 11
U920(0) → 12
U620(0) → 13
U410(0, 0) → 14
plus0(0, 0) → 15
U420(0) → 16
U1030(0, 0, 0) → 17
proper0(0) → 18
isNatKind0(0) → 19
U820(0, 0, 0) → 20
U110(0, 0, 0) → 21
U510(0) → 22
U150(0, 0) → 23
x0(0, 0) → 24
U320(0, 0, 0) → 25
U130(0, 0, 0) → 26
U160(0) → 27
U210(0, 0) → 28
s0(0) → 29
U230(0) → 30
U340(0, 0, 0) → 31
U710(0, 0) → 32
U720(0, 0) → 33
U120(0, 0, 0) → 34
U830(0, 0, 0) → 35
U220(0, 0) → 36
U1020(0, 0, 0) → 37
U140(0, 0, 0) → 38
U330(0, 0, 0) → 39
U1041(0, 0, 0) → 40
ok1(40) → 1
active1(0) → 41
top1(41) → 2
U351(0, 0) → 42
ok1(42) → 3
U811(0, 0, 0) → 43
mark1(43) → 4
U311(0, 0, 0) → 44
ok1(44) → 5
U911(0, 0) → 45
mark1(45) → 6
U1041(0, 0, 0) → 46
mark1(46) → 1
U811(0, 0, 0) → 47
ok1(47) → 4
U361(0) → 48
mark1(48) → 7
isNat1(0) → 49
ok1(49) → 8
U1011(0, 0, 0) → 50
ok1(50) → 9
U611(0, 0) → 51
ok1(51) → 10
U841(0, 0, 0) → 52
mark1(52) → 11
U921(0) → 53
mark1(53) → 12
U621(0) → 54
mark1(54) → 13
U351(0, 0) → 55
mark1(55) → 3
U411(0, 0) → 56
mark1(56) → 14
plus1(0, 0) → 57
ok1(57) → 15
plus1(0, 0) → 58
mark1(58) → 15
U421(0) → 59
mark1(59) → 16
U311(0, 0, 0) → 60
mark1(60) → 5
U1031(0, 0, 0) → 61
mark1(61) → 17
tt1() → 62
ok1(62) → 18
isNatKind1(0) → 63
ok1(63) → 19
U821(0, 0, 0) → 64
ok1(64) → 20
U111(0, 0, 0) → 65
mark1(65) → 21
U821(0, 0, 0) → 66
mark1(66) → 20
U511(0) → 67
mark1(67) → 22
U111(0, 0, 0) → 68
ok1(68) → 21
U151(0, 0) → 69
ok1(69) → 23
U611(0, 0) → 70
mark1(70) → 10
U841(0, 0, 0) → 71
ok1(71) → 11
x1(0, 0) → 72
ok1(72) → 24
U321(0, 0, 0) → 73
ok1(73) → 25
U131(0, 0, 0) → 74
mark1(74) → 26
U161(0) → 75
mark1(75) → 27
U211(0, 0) → 76
ok1(76) → 28
U361(0) → 77
ok1(77) → 7
s1(0) → 78
ok1(78) → 29
U231(0) → 79
ok1(79) → 30
U1011(0, 0, 0) → 80
mark1(80) → 9
U341(0, 0, 0) → 81
ok1(81) → 31
U711(0, 0) → 82
ok1(82) → 32
01() → 83
ok1(83) → 18
U131(0, 0, 0) → 84
ok1(84) → 26
U341(0, 0, 0) → 85
mark1(85) → 31
U721(0, 0) → 86
mark1(86) → 33
U121(0, 0, 0) → 87
ok1(87) → 34
U831(0, 0, 0) → 88
ok1(88) → 35
U221(0, 0) → 89
mark1(89) → 36
U1021(0, 0, 0) → 90
mark1(90) → 37
U121(0, 0, 0) → 91
mark1(91) → 34
U921(0) → 92
ok1(92) → 12
U621(0) → 93
ok1(93) → 13
U721(0, 0) → 94
ok1(94) → 33
U321(0, 0, 0) → 95
mark1(95) → 25
U1021(0, 0, 0) → 96
ok1(96) → 37
U1031(0, 0, 0) → 97
ok1(97) → 17
U421(0) → 98
ok1(98) → 16
U211(0, 0) → 99
mark1(99) → 28
x1(0, 0) → 100
mark1(100) → 24
U411(0, 0) → 101
ok1(101) → 14
U711(0, 0) → 102
mark1(102) → 32
U231(0) → 103
mark1(103) → 30
U141(0, 0, 0) → 104
ok1(104) → 38
U151(0, 0) → 105
mark1(105) → 23
U161(0) → 106
ok1(106) → 27
U331(0, 0, 0) → 107
mark1(107) → 39
U331(0, 0, 0) → 108
ok1(108) → 39
U511(0) → 109
ok1(109) → 22
s1(0) → 110
mark1(110) → 29
U141(0, 0, 0) → 111
mark1(111) → 38
U221(0, 0) → 112
ok1(112) → 36
U831(0, 0, 0) → 113
mark1(113) → 35
proper1(0) → 114
top1(114) → 2
U911(0, 0) → 115
ok1(115) → 6
ok1(40) → 40
ok1(40) → 46
ok1(42) → 42
ok1(42) → 55
mark1(43) → 43
mark1(43) → 47
ok1(44) → 44
ok1(44) → 60
mark1(45) → 45
mark1(45) → 115
mark1(46) → 40
mark1(46) → 46
ok1(47) → 43
ok1(47) → 47
mark1(48) → 48
mark1(48) → 77
ok1(49) → 49
ok1(50) → 50
ok1(50) → 80
ok1(51) → 51
ok1(51) → 70
mark1(52) → 52
mark1(52) → 71
mark1(53) → 53
mark1(53) → 92
mark1(54) → 54
mark1(54) → 93
mark1(55) → 42
mark1(55) → 55
mark1(56) → 56
mark1(56) → 101
ok1(57) → 57
ok1(57) → 58
mark1(58) → 57
mark1(58) → 58
mark1(59) → 59
mark1(59) → 98
mark1(60) → 44
mark1(60) → 60
mark1(61) → 61
mark1(61) → 97
ok1(62) → 114
ok1(63) → 63
ok1(64) → 64
ok1(64) → 66
mark1(65) → 65
mark1(65) → 68
mark1(66) → 64
mark1(66) → 66
mark1(67) → 67
mark1(67) → 109
ok1(68) → 65
ok1(68) → 68
ok1(69) → 69
ok1(69) → 105
mark1(70) → 51
mark1(70) → 70
ok1(71) → 52
ok1(71) → 71
ok1(72) → 72
ok1(72) → 100
ok1(73) → 73
ok1(73) → 95
mark1(74) → 74
mark1(74) → 84
mark1(75) → 75
mark1(75) → 106
ok1(76) → 76
ok1(76) → 99
ok1(77) → 48
ok1(77) → 77
ok1(78) → 78
ok1(78) → 110
ok1(79) → 79
ok1(79) → 103
mark1(80) → 50
mark1(80) → 80
ok1(81) → 81
ok1(81) → 85
ok1(82) → 82
ok1(82) → 102
ok1(83) → 114
ok1(84) → 74
ok1(84) → 84
mark1(85) → 81
mark1(85) → 85
mark1(86) → 86
mark1(86) → 94
ok1(87) → 87
ok1(87) → 91
ok1(88) → 88
ok1(88) → 113
mark1(89) → 89
mark1(89) → 112
mark1(90) → 90
mark1(90) → 96
mark1(91) → 87
mark1(91) → 91
ok1(92) → 53
ok1(92) → 92
ok1(93) → 54
ok1(93) → 93
ok1(94) → 86
ok1(94) → 94
mark1(95) → 73
mark1(95) → 95
ok1(96) → 90
ok1(96) → 96
ok1(97) → 61
ok1(97) → 97
ok1(98) → 59
ok1(98) → 98
mark1(99) → 76
mark1(99) → 99
mark1(100) → 72
mark1(100) → 100
ok1(101) → 56
ok1(101) → 101
mark1(102) → 82
mark1(102) → 102
mark1(103) → 79
mark1(103) → 103
ok1(104) → 104
ok1(104) → 111
mark1(105) → 69
mark1(105) → 105
ok1(106) → 75
ok1(106) → 106
mark1(107) → 107
mark1(107) → 108
ok1(108) → 107
ok1(108) → 108
ok1(109) → 67
ok1(109) → 109
mark1(110) → 78
mark1(110) → 110
mark1(111) → 104
mark1(111) → 111
ok1(112) → 89
ok1(112) → 112
mark1(113) → 88
mark1(113) → 113
ok1(115) → 45
ok1(115) → 115
active2(62) → 116
top2(116) → 2
active2(83) → 116

(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:

U104(ok(z0), ok(z1), ok(z2)) → ok(U104(z0, z1, z2))
U104(mark(z0), z1, z2) → mark(U104(z0, z1, z2))
top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
U35(ok(z0), ok(z1)) → ok(U35(z0, z1))
U35(mark(z0), z1) → mark(U35(z0, z1))
U81(mark(z0), z1, z2) → mark(U81(z0, z1, z2))
U81(ok(z0), ok(z1), ok(z2)) → ok(U81(z0, z1, z2))
U31(ok(z0), ok(z1), ok(z2)) → ok(U31(z0, z1, z2))
U31(mark(z0), z1, z2) → mark(U31(z0, z1, z2))
U91(mark(z0), z1) → mark(U91(z0, z1))
U91(ok(z0), ok(z1)) → ok(U91(z0, z1))
U36(mark(z0)) → mark(U36(z0))
U36(ok(z0)) → ok(U36(z0))
isNat(ok(z0)) → ok(isNat(z0))
U101(ok(z0), ok(z1), ok(z2)) → ok(U101(z0, z1, z2))
U101(mark(z0), z1, z2) → mark(U101(z0, z1, z2))
U61(ok(z0), ok(z1)) → ok(U61(z0, z1))
U61(mark(z0), z1) → mark(U61(z0, z1))
U84(mark(z0), z1, z2) → mark(U84(z0, z1, z2))
U84(ok(z0), ok(z1), ok(z2)) → ok(U84(z0, z1, z2))
U92(mark(z0)) → mark(U92(z0))
U92(ok(z0)) → ok(U92(z0))
U62(mark(z0)) → mark(U62(z0))
U62(ok(z0)) → ok(U62(z0))
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))
U42(mark(z0)) → mark(U42(z0))
U42(ok(z0)) → ok(U42(z0))
U103(mark(z0), z1, z2) → mark(U103(z0, z1, z2))
U103(ok(z0), ok(z1), ok(z2)) → ok(U103(z0, z1, z2))
proper(tt) → ok(tt)
proper(0) → ok(0)
isNatKind(ok(z0)) → ok(isNatKind(z0))
U82(ok(z0), ok(z1), ok(z2)) → ok(U82(z0, z1, z2))
U82(mark(z0), z1, z2) → mark(U82(z0, z1, z2))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
U51(mark(z0)) → mark(U51(z0))
U51(ok(z0)) → ok(U51(z0))
U15(ok(z0), ok(z1)) → ok(U15(z0, z1))
U15(mark(z0), z1) → mark(U15(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
U32(ok(z0), ok(z1), ok(z2)) → ok(U32(z0, z1, z2))
U32(mark(z0), z1, z2) → mark(U32(z0, z1, z2))
U13(mark(z0), z1, z2) → mark(U13(z0, z1, z2))
U13(ok(z0), ok(z1), ok(z2)) → ok(U13(z0, z1, z2))
U16(mark(z0)) → mark(U16(z0))
U16(ok(z0)) → ok(U16(z0))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
s(ok(z0)) → ok(s(z0))
s(mark(z0)) → mark(s(z0))
U23(ok(z0)) → ok(U23(z0))
U23(mark(z0)) → mark(U23(z0))
U34(ok(z0), ok(z1), ok(z2)) → ok(U34(z0, z1, z2))
U34(mark(z0), z1, z2) → mark(U34(z0, z1, z2))
U71(ok(z0), ok(z1)) → ok(U71(z0, z1))
U71(mark(z0), z1) → mark(U71(z0, z1))
U72(mark(z0), z1) → mark(U72(z0, z1))
U72(ok(z0), ok(z1)) → ok(U72(z0, z1))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
U12(mark(z0), z1, z2) → mark(U12(z0, z1, z2))
U83(ok(z0), ok(z1), ok(z2)) → ok(U83(z0, z1, z2))
U83(mark(z0), z1, z2) → mark(U83(z0, z1, z2))
U22(mark(z0), z1) → mark(U22(z0, z1))
U22(ok(z0), ok(z1)) → ok(U22(z0, z1))
U102(mark(z0), z1, z2) → mark(U102(z0, z1, z2))
U102(ok(z0), ok(z1), ok(z2)) → ok(U102(z0, z1, z2))
U14(ok(z0), ok(z1), ok(z2)) → ok(U14(z0, z1, z2))
U14(mark(z0), z1, z2) → mark(U14(z0, z1, z2))
U33(mark(z0), z1, z2) → mark(U33(z0, z1, z2))
U33(ok(z0), ok(z1), ok(z2)) → ok(U33(z0, z1, z2))

Tuples:

U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
TOP(ok(z0)) → c2(TOP(active(z0)))
TOP(mark(z0)) → c3(TOP(proper(z0)), PROPER(z0))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
U92'(mark(z0)) → c21(U92'(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(mark(z0)) → c23(U62'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U42'(mark(z0)) → c30(U42'(z0))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
PROPER(tt) → c34
PROPER(0) → c35
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
U51'(ok(z0)) → c42(U51'(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
X(z0, mark(z1)) → c47(X(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
U16'(ok(z0)) → c53(U16'(z0))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
S(ok(z0)) → c56(S(z0))
S(mark(z0)) → c57(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))

S tuples:

U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
TOP(ok(z0)) → c2(TOP(active(z0)))
TOP(mark(z0)) → c3(TOP(proper(z0)), PROPER(z0))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
U92'(mark(z0)) → c21(U92'(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(mark(z0)) → c23(U62'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U42'(mark(z0)) → c30(U42'(z0))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
PROPER(tt) → c34
PROPER(0) → c35
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
U51'(ok(z0)) → c42(U51'(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
X(z0, mark(z1)) → c47(X(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
U16'(ok(z0)) → c53(U16'(z0))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
S(ok(z0)) → c56(S(z0))
S(mark(z0)) → c57(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))

K tuples:none
Defined Rule Symbols:

U104, top, U35, U81, U31, U91, U36, isNat, U101, U61, U84, U92, U62, U41, plus, U42, U103, proper, isNatKind, U82, U11, U51, U15, x, U32, U13, U16, U21, s, U23, U34, U71, U72, U12, U83, U22, U102, U14, U33

Defined Pair Symbols:

U104', TOP, U35', U81', U31', U91', U36', ISNAT, U101', U61', U84', U92', U62', U41', PLUS, U42', U103', PROPER, ISNATKIND, U82', U11', U51', U15', X, U32', U13', U16', U21', S, U23', U34', U71', U72', U12', U83', U22', U102', U14', U33'

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c70, c71, c72, c73, c74, c75, c76, c77

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

Removed 3 trailing nodes:

PROPER(tt) → c34
PROPER(0) → c35
TOP(ok(z0)) → c2(TOP(active(z0)))

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

U104(ok(z0), ok(z1), ok(z2)) → ok(U104(z0, z1, z2))
U104(mark(z0), z1, z2) → mark(U104(z0, z1, z2))
top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
U35(ok(z0), ok(z1)) → ok(U35(z0, z1))
U35(mark(z0), z1) → mark(U35(z0, z1))
U81(mark(z0), z1, z2) → mark(U81(z0, z1, z2))
U81(ok(z0), ok(z1), ok(z2)) → ok(U81(z0, z1, z2))
U31(ok(z0), ok(z1), ok(z2)) → ok(U31(z0, z1, z2))
U31(mark(z0), z1, z2) → mark(U31(z0, z1, z2))
U91(mark(z0), z1) → mark(U91(z0, z1))
U91(ok(z0), ok(z1)) → ok(U91(z0, z1))
U36(mark(z0)) → mark(U36(z0))
U36(ok(z0)) → ok(U36(z0))
isNat(ok(z0)) → ok(isNat(z0))
U101(ok(z0), ok(z1), ok(z2)) → ok(U101(z0, z1, z2))
U101(mark(z0), z1, z2) → mark(U101(z0, z1, z2))
U61(ok(z0), ok(z1)) → ok(U61(z0, z1))
U61(mark(z0), z1) → mark(U61(z0, z1))
U84(mark(z0), z1, z2) → mark(U84(z0, z1, z2))
U84(ok(z0), ok(z1), ok(z2)) → ok(U84(z0, z1, z2))
U92(mark(z0)) → mark(U92(z0))
U92(ok(z0)) → ok(U92(z0))
U62(mark(z0)) → mark(U62(z0))
U62(ok(z0)) → ok(U62(z0))
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))
U42(mark(z0)) → mark(U42(z0))
U42(ok(z0)) → ok(U42(z0))
U103(mark(z0), z1, z2) → mark(U103(z0, z1, z2))
U103(ok(z0), ok(z1), ok(z2)) → ok(U103(z0, z1, z2))
proper(tt) → ok(tt)
proper(0) → ok(0)
isNatKind(ok(z0)) → ok(isNatKind(z0))
U82(ok(z0), ok(z1), ok(z2)) → ok(U82(z0, z1, z2))
U82(mark(z0), z1, z2) → mark(U82(z0, z1, z2))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
U51(mark(z0)) → mark(U51(z0))
U51(ok(z0)) → ok(U51(z0))
U15(ok(z0), ok(z1)) → ok(U15(z0, z1))
U15(mark(z0), z1) → mark(U15(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
U32(ok(z0), ok(z1), ok(z2)) → ok(U32(z0, z1, z2))
U32(mark(z0), z1, z2) → mark(U32(z0, z1, z2))
U13(mark(z0), z1, z2) → mark(U13(z0, z1, z2))
U13(ok(z0), ok(z1), ok(z2)) → ok(U13(z0, z1, z2))
U16(mark(z0)) → mark(U16(z0))
U16(ok(z0)) → ok(U16(z0))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
s(ok(z0)) → ok(s(z0))
s(mark(z0)) → mark(s(z0))
U23(ok(z0)) → ok(U23(z0))
U23(mark(z0)) → mark(U23(z0))
U34(ok(z0), ok(z1), ok(z2)) → ok(U34(z0, z1, z2))
U34(mark(z0), z1, z2) → mark(U34(z0, z1, z2))
U71(ok(z0), ok(z1)) → ok(U71(z0, z1))
U71(mark(z0), z1) → mark(U71(z0, z1))
U72(mark(z0), z1) → mark(U72(z0, z1))
U72(ok(z0), ok(z1)) → ok(U72(z0, z1))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
U12(mark(z0), z1, z2) → mark(U12(z0, z1, z2))
U83(ok(z0), ok(z1), ok(z2)) → ok(U83(z0, z1, z2))
U83(mark(z0), z1, z2) → mark(U83(z0, z1, z2))
U22(mark(z0), z1) → mark(U22(z0, z1))
U22(ok(z0), ok(z1)) → ok(U22(z0, z1))
U102(mark(z0), z1, z2) → mark(U102(z0, z1, z2))
U102(ok(z0), ok(z1), ok(z2)) → ok(U102(z0, z1, z2))
U14(ok(z0), ok(z1), ok(z2)) → ok(U14(z0, z1, z2))
U14(mark(z0), z1, z2) → mark(U14(z0, z1, z2))
U33(mark(z0), z1, z2) → mark(U33(z0, z1, z2))
U33(ok(z0), ok(z1), ok(z2)) → ok(U33(z0, z1, z2))

Tuples:

U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)), PROPER(z0))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
U92'(mark(z0)) → c21(U92'(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(mark(z0)) → c23(U62'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U42'(mark(z0)) → c30(U42'(z0))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
U51'(ok(z0)) → c42(U51'(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
X(z0, mark(z1)) → c47(X(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
U16'(ok(z0)) → c53(U16'(z0))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
S(ok(z0)) → c56(S(z0))
S(mark(z0)) → c57(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))

S tuples:

U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)), PROPER(z0))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
U92'(mark(z0)) → c21(U92'(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(mark(z0)) → c23(U62'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U42'(mark(z0)) → c30(U42'(z0))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
U51'(ok(z0)) → c42(U51'(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
X(z0, mark(z1)) → c47(X(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
U16'(ok(z0)) → c53(U16'(z0))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
S(ok(z0)) → c56(S(z0))
S(mark(z0)) → c57(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))

K tuples:none
Defined Rule Symbols:

U104, top, U35, U81, U31, U91, U36, isNat, U101, U61, U84, U92, U62, U41, plus, U42, U103, proper, isNatKind, U82, U11, U51, U15, x, U32, U13, U16, U21, s, U23, U34, U71, U72, U12, U83, U22, U102, U14, U33

Defined Pair Symbols:

U104', TOP, U35', U81', U31', U91', U36', ISNAT, U101', U61', U84', U92', U62', U41', PLUS, U42', U103', ISNATKIND, U82', U11', U51', U15', X, U32', U13', U16', U21', S, U23', U34', U71', U72', U12', U83', U22', U102', U14', U33'

Compound Symbols:

c, c1, 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, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c70, c71, c72, c73, c74, c75, c76, c77

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

Removed 1 trailing tuple parts

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

U104(ok(z0), ok(z1), ok(z2)) → ok(U104(z0, z1, z2))
U104(mark(z0), z1, z2) → mark(U104(z0, z1, z2))
top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
U35(ok(z0), ok(z1)) → ok(U35(z0, z1))
U35(mark(z0), z1) → mark(U35(z0, z1))
U81(mark(z0), z1, z2) → mark(U81(z0, z1, z2))
U81(ok(z0), ok(z1), ok(z2)) → ok(U81(z0, z1, z2))
U31(ok(z0), ok(z1), ok(z2)) → ok(U31(z0, z1, z2))
U31(mark(z0), z1, z2) → mark(U31(z0, z1, z2))
U91(mark(z0), z1) → mark(U91(z0, z1))
U91(ok(z0), ok(z1)) → ok(U91(z0, z1))
U36(mark(z0)) → mark(U36(z0))
U36(ok(z0)) → ok(U36(z0))
isNat(ok(z0)) → ok(isNat(z0))
U101(ok(z0), ok(z1), ok(z2)) → ok(U101(z0, z1, z2))
U101(mark(z0), z1, z2) → mark(U101(z0, z1, z2))
U61(ok(z0), ok(z1)) → ok(U61(z0, z1))
U61(mark(z0), z1) → mark(U61(z0, z1))
U84(mark(z0), z1, z2) → mark(U84(z0, z1, z2))
U84(ok(z0), ok(z1), ok(z2)) → ok(U84(z0, z1, z2))
U92(mark(z0)) → mark(U92(z0))
U92(ok(z0)) → ok(U92(z0))
U62(mark(z0)) → mark(U62(z0))
U62(ok(z0)) → ok(U62(z0))
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))
U42(mark(z0)) → mark(U42(z0))
U42(ok(z0)) → ok(U42(z0))
U103(mark(z0), z1, z2) → mark(U103(z0, z1, z2))
U103(ok(z0), ok(z1), ok(z2)) → ok(U103(z0, z1, z2))
proper(tt) → ok(tt)
proper(0) → ok(0)
isNatKind(ok(z0)) → ok(isNatKind(z0))
U82(ok(z0), ok(z1), ok(z2)) → ok(U82(z0, z1, z2))
U82(mark(z0), z1, z2) → mark(U82(z0, z1, z2))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
U51(mark(z0)) → mark(U51(z0))
U51(ok(z0)) → ok(U51(z0))
U15(ok(z0), ok(z1)) → ok(U15(z0, z1))
U15(mark(z0), z1) → mark(U15(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
U32(ok(z0), ok(z1), ok(z2)) → ok(U32(z0, z1, z2))
U32(mark(z0), z1, z2) → mark(U32(z0, z1, z2))
U13(mark(z0), z1, z2) → mark(U13(z0, z1, z2))
U13(ok(z0), ok(z1), ok(z2)) → ok(U13(z0, z1, z2))
U16(mark(z0)) → mark(U16(z0))
U16(ok(z0)) → ok(U16(z0))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
s(ok(z0)) → ok(s(z0))
s(mark(z0)) → mark(s(z0))
U23(ok(z0)) → ok(U23(z0))
U23(mark(z0)) → mark(U23(z0))
U34(ok(z0), ok(z1), ok(z2)) → ok(U34(z0, z1, z2))
U34(mark(z0), z1, z2) → mark(U34(z0, z1, z2))
U71(ok(z0), ok(z1)) → ok(U71(z0, z1))
U71(mark(z0), z1) → mark(U71(z0, z1))
U72(mark(z0), z1) → mark(U72(z0, z1))
U72(ok(z0), ok(z1)) → ok(U72(z0, z1))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
U12(mark(z0), z1, z2) → mark(U12(z0, z1, z2))
U83(ok(z0), ok(z1), ok(z2)) → ok(U83(z0, z1, z2))
U83(mark(z0), z1, z2) → mark(U83(z0, z1, z2))
U22(mark(z0), z1) → mark(U22(z0, z1))
U22(ok(z0), ok(z1)) → ok(U22(z0, z1))
U102(mark(z0), z1, z2) → mark(U102(z0, z1, z2))
U102(ok(z0), ok(z1), ok(z2)) → ok(U102(z0, z1, z2))
U14(ok(z0), ok(z1), ok(z2)) → ok(U14(z0, z1, z2))
U14(mark(z0), z1, z2) → mark(U14(z0, z1, z2))
U33(mark(z0), z1, z2) → mark(U33(z0, z1, z2))
U33(ok(z0), ok(z1), ok(z2)) → ok(U33(z0, z1, z2))

Tuples:

U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
U92'(mark(z0)) → c21(U92'(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(mark(z0)) → c23(U62'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U42'(mark(z0)) → c30(U42'(z0))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
U51'(ok(z0)) → c42(U51'(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
X(z0, mark(z1)) → c47(X(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
U16'(ok(z0)) → c53(U16'(z0))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
S(ok(z0)) → c56(S(z0))
S(mark(z0)) → c57(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
U92'(mark(z0)) → c21(U92'(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(mark(z0)) → c23(U62'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U42'(mark(z0)) → c30(U42'(z0))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
U51'(ok(z0)) → c42(U51'(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
X(z0, mark(z1)) → c47(X(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
U16'(ok(z0)) → c53(U16'(z0))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
S(ok(z0)) → c56(S(z0))
S(mark(z0)) → c57(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

K tuples:none
Defined Rule Symbols:

U104, top, U35, U81, U31, U91, U36, isNat, U101, U61, U84, U92, U62, U41, plus, U42, U103, proper, isNatKind, U82, U11, U51, U15, x, U32, U13, U16, U21, s, U23, U34, U71, U72, U12, U83, U22, U102, U14, U33

Defined Pair Symbols:

U104', U35', U81', U31', U91', U36', ISNAT, U101', U61', U84', U92', U62', U41', PLUS, U42', U103', ISNATKIND, U82', U11', U51', U15', X, U32', U13', U16', U21', S, U23', U34', U71', U72', U12', U83', U22', U102', U14', U33', TOP

Compound Symbols:

c, c1, 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, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c70, c71, c72, c73, c74, c75, c76, c77, c3

(11) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

U104(ok(z0), ok(z1), ok(z2)) → ok(U104(z0, z1, z2))
U104(mark(z0), z1, z2) → mark(U104(z0, z1, z2))
top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
U35(ok(z0), ok(z1)) → ok(U35(z0, z1))
U35(mark(z0), z1) → mark(U35(z0, z1))
U81(mark(z0), z1, z2) → mark(U81(z0, z1, z2))
U81(ok(z0), ok(z1), ok(z2)) → ok(U81(z0, z1, z2))
U31(ok(z0), ok(z1), ok(z2)) → ok(U31(z0, z1, z2))
U31(mark(z0), z1, z2) → mark(U31(z0, z1, z2))
U91(mark(z0), z1) → mark(U91(z0, z1))
U91(ok(z0), ok(z1)) → ok(U91(z0, z1))
U36(mark(z0)) → mark(U36(z0))
U36(ok(z0)) → ok(U36(z0))
isNat(ok(z0)) → ok(isNat(z0))
U101(ok(z0), ok(z1), ok(z2)) → ok(U101(z0, z1, z2))
U101(mark(z0), z1, z2) → mark(U101(z0, z1, z2))
U61(ok(z0), ok(z1)) → ok(U61(z0, z1))
U61(mark(z0), z1) → mark(U61(z0, z1))
U84(mark(z0), z1, z2) → mark(U84(z0, z1, z2))
U84(ok(z0), ok(z1), ok(z2)) → ok(U84(z0, z1, z2))
U92(mark(z0)) → mark(U92(z0))
U92(ok(z0)) → ok(U92(z0))
U62(mark(z0)) → mark(U62(z0))
U62(ok(z0)) → ok(U62(z0))
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))
U42(mark(z0)) → mark(U42(z0))
U42(ok(z0)) → ok(U42(z0))
U103(mark(z0), z1, z2) → mark(U103(z0, z1, z2))
U103(ok(z0), ok(z1), ok(z2)) → ok(U103(z0, z1, z2))
isNatKind(ok(z0)) → ok(isNatKind(z0))
U82(ok(z0), ok(z1), ok(z2)) → ok(U82(z0, z1, z2))
U82(mark(z0), z1, z2) → mark(U82(z0, z1, z2))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
U51(mark(z0)) → mark(U51(z0))
U51(ok(z0)) → ok(U51(z0))
U15(ok(z0), ok(z1)) → ok(U15(z0, z1))
U15(mark(z0), z1) → mark(U15(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
U32(ok(z0), ok(z1), ok(z2)) → ok(U32(z0, z1, z2))
U32(mark(z0), z1, z2) → mark(U32(z0, z1, z2))
U13(mark(z0), z1, z2) → mark(U13(z0, z1, z2))
U13(ok(z0), ok(z1), ok(z2)) → ok(U13(z0, z1, z2))
U16(mark(z0)) → mark(U16(z0))
U16(ok(z0)) → ok(U16(z0))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
s(ok(z0)) → ok(s(z0))
s(mark(z0)) → mark(s(z0))
U23(ok(z0)) → ok(U23(z0))
U23(mark(z0)) → mark(U23(z0))
U34(ok(z0), ok(z1), ok(z2)) → ok(U34(z0, z1, z2))
U34(mark(z0), z1, z2) → mark(U34(z0, z1, z2))
U71(ok(z0), ok(z1)) → ok(U71(z0, z1))
U71(mark(z0), z1) → mark(U71(z0, z1))
U72(mark(z0), z1) → mark(U72(z0, z1))
U72(ok(z0), ok(z1)) → ok(U72(z0, z1))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
U12(mark(z0), z1, z2) → mark(U12(z0, z1, z2))
U83(ok(z0), ok(z1), ok(z2)) → ok(U83(z0, z1, z2))
U83(mark(z0), z1, z2) → mark(U83(z0, z1, z2))
U22(mark(z0), z1) → mark(U22(z0, z1))
U22(ok(z0), ok(z1)) → ok(U22(z0, z1))
U102(mark(z0), z1, z2) → mark(U102(z0, z1, z2))
U102(ok(z0), ok(z1), ok(z2)) → ok(U102(z0, z1, z2))
U14(ok(z0), ok(z1), ok(z2)) → ok(U14(z0, z1, z2))
U14(mark(z0), z1, z2) → mark(U14(z0, z1, z2))
U33(mark(z0), z1, z2) → mark(U33(z0, z1, z2))
U33(ok(z0), ok(z1), ok(z2)) → ok(U33(z0, z1, z2))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)

Tuples:

U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
U92'(mark(z0)) → c21(U92'(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(mark(z0)) → c23(U62'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U42'(mark(z0)) → c30(U42'(z0))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
U51'(ok(z0)) → c42(U51'(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
X(z0, mark(z1)) → c47(X(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
U16'(ok(z0)) → c53(U16'(z0))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
S(ok(z0)) → c56(S(z0))
S(mark(z0)) → c57(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
U92'(mark(z0)) → c21(U92'(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(mark(z0)) → c23(U62'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U42'(mark(z0)) → c30(U42'(z0))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
U51'(ok(z0)) → c42(U51'(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
X(z0, mark(z1)) → c47(X(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
U16'(ok(z0)) → c53(U16'(z0))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
S(ok(z0)) → c56(S(z0))
S(mark(z0)) → c57(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

K tuples:none
Defined Rule Symbols:

proper

Defined Pair Symbols:

U104', U35', U81', U31', U91', U36', ISNAT, U101', U61', U84', U92', U62', U41', PLUS, U42', U103', ISNATKIND, U82', U11', U51', U15', X, U32', U13', U16', U21', S, U23', U34', U71', U72', U12', U83', U22', U102', U14', U33', TOP

Compound Symbols:

c, c1, 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, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c70, c71, c72, c73, c74, c75, c76, c77, c3

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

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

TOP(mark(z0)) → c3(TOP(proper(z0)))

We considered the (Usable) Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)

And the Tuples:

U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
U92'(mark(z0)) → c21(U92'(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(mark(z0)) → c23(U62'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U42'(mark(z0)) → c30(U42'(z0))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
U51'(ok(z0)) → c42(U51'(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
X(z0, mark(z1)) → c47(X(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
U16'(ok(z0)) → c53(U16'(z0))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
S(ok(z0)) → c56(S(z0))
S(mark(z0)) → c57(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

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

POL(0) = 0
POL(ISNAT(x₁)) = 0
POL(ISNATKIND(x₁)) = 0
POL(PLUS(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(TOP(x₁)) = x₁
POL(U101'(x₁, x₂, x₃)) = 0
POL(U102'(x₁, x₂, x₃)) = 0
POL(U103'(x₁, x₂, x₃)) = 0
POL(U104'(x₁, x₂, x₃)) = 0
POL(U11'(x₁, x₂, x₃)) = 0
POL(U12'(x₁, x₂, x₃)) = 0
POL(U13'(x₁, x₂, x₃)) = 0
POL(U14'(x₁, x₂, x₃)) = 0
POL(U15'(x₁, x₂)) = 0
POL(U16'(x₁)) = 0
POL(U21'(x₁, x₂)) = 0
POL(U22'(x₁, x₂)) = 0
POL(U23'(x₁)) = 0
POL(U31'(x₁, x₂, x₃)) = 0
POL(U32'(x₁, x₂, x₃)) = 0
POL(U33'(x₁, x₂, x₃)) = 0
POL(U34'(x₁, x₂, x₃)) = 0
POL(U35'(x₁, x₂)) = 0
POL(U36'(x₁)) = 0
POL(U41'(x₁, x₂)) = 0
POL(U42'(x₁)) = 0
POL(U51'(x₁)) = 0
POL(U61'(x₁, x₂)) = 0
POL(U62'(x₁)) = 0
POL(U71'(x₁, x₂)) = 0
POL(U72'(x₁, x₂)) = 0
POL(U81'(x₁, x₂, x₃)) = 0
POL(U82'(x₁, x₂, x₃)) = 0
POL(U83'(x₁, x₂, x₃)) = 0
POL(U84'(x₁, x₂, x₃)) = 0
POL(U91'(x₁, x₂)) = 0
POL(U92'(x₁)) = 0
POL(X(x₁, x₂)) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(c41(x₁)) = x₁
POL(c42(x₁)) = x₁
POL(c43(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(c45(x₁)) = x₁
POL(c46(x₁)) = x₁
POL(c47(x₁)) = x₁
POL(c48(x₁)) = x₁
POL(c49(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c50(x₁)) = x₁
POL(c51(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(c53(x₁)) = x₁
POL(c54(x₁)) = x₁
POL(c55(x₁)) = x₁
POL(c56(x₁)) = x₁
POL(c57(x₁)) = x₁
POL(c58(x₁)) = x₁
POL(c59(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c60(x₁)) = x₁
POL(c61(x₁)) = x₁
POL(c62(x₁)) = x₁
POL(c63(x₁)) = x₁
POL(c64(x₁)) = x₁
POL(c65(x₁)) = x₁
POL(c66(x₁)) = x₁
POL(c67(x₁)) = x₁
POL(c68(x₁)) = x₁
POL(c69(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c70(x₁)) = x₁
POL(c71(x₁)) = x₁
POL(c72(x₁)) = x₁
POL(c73(x₁)) = x₁
POL(c74(x₁)) = x₁
POL(c75(x₁)) = x₁
POL(c76(x₁)) = x₁
POL(c77(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

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)

Tuples:

U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
U92'(mark(z0)) → c21(U92'(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(mark(z0)) → c23(U62'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U42'(mark(z0)) → c30(U42'(z0))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
U51'(ok(z0)) → c42(U51'(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
X(z0, mark(z1)) → c47(X(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
U16'(ok(z0)) → c53(U16'(z0))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
S(ok(z0)) → c56(S(z0))
S(mark(z0)) → c57(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
U92'(mark(z0)) → c21(U92'(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(mark(z0)) → c23(U62'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U42'(mark(z0)) → c30(U42'(z0))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
U51'(ok(z0)) → c42(U51'(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
X(z0, mark(z1)) → c47(X(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
U16'(ok(z0)) → c53(U16'(z0))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
S(ok(z0)) → c56(S(z0))
S(mark(z0)) → c57(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U104', U35', U81', U31', U91', U36', ISNAT, U101', U61', U84', U92', U62', U41', PLUS, U42', U103', ISNATKIND, U82', U11', U51', U15', X, U32', U13', U16', U21', S, U23', U34', U71', U72', U12', U83', U22', U102', U14', U33', TOP

Compound Symbols:

c, c1, 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, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c70, c71, c72, c73, c74, c75, c76, c77, c3

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

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

U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))

We considered the (Usable) Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)

And the Tuples:

U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
U92'(mark(z0)) → c21(U92'(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(mark(z0)) → c23(U62'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U42'(mark(z0)) → c30(U42'(z0))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
U51'(ok(z0)) → c42(U51'(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
X(z0, mark(z1)) → c47(X(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
U16'(ok(z0)) → c53(U16'(z0))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
S(ok(z0)) → c56(S(z0))
S(mark(z0)) → c57(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

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

POL(0) = 0
POL(ISNAT(x₁)) = 0
POL(ISNATKIND(x₁)) = 0
POL(PLUS(x₁, x₂)) = x₂
POL(S(x₁)) = 0
POL(TOP(x₁)) = x₁
POL(U101'(x₁, x₂, x₃)) = 0
POL(U102'(x₁, x₂, x₃)) = 0
POL(U103'(x₁, x₂, x₃)) = 0
POL(U104'(x₁, x₂, x₃)) = 0
POL(U11'(x₁, x₂, x₃)) = 0
POL(U12'(x₁, x₂, x₃)) = 0
POL(U13'(x₁, x₂, x₃)) = 0
POL(U14'(x₁, x₂, x₃)) = 0
POL(U15'(x₁, x₂)) = 0
POL(U16'(x₁)) = 0
POL(U21'(x₁, x₂)) = x₁
POL(U22'(x₁, x₂)) = 0
POL(U23'(x₁)) = 0
POL(U31'(x₁, x₂, x₃)) = x₁ + x₂
POL(U32'(x₁, x₂, x₃)) = x₃
POL(U33'(x₁, x₂, x₃)) = x₁
POL(U34'(x₁, x₂, x₃)) = x₁
POL(U35'(x₁, x₂)) = 0
POL(U36'(x₁)) = x₁
POL(U41'(x₁, x₂)) = 0
POL(U42'(x₁)) = 0
POL(U51'(x₁)) = 0
POL(U61'(x₁, x₂)) = 0
POL(U62'(x₁)) = 0
POL(U71'(x₁, x₂)) = 0
POL(U72'(x₁, x₂)) = 0
POL(U81'(x₁, x₂, x₃)) = 0
POL(U82'(x₁, x₂, x₃)) = 0
POL(U83'(x₁, x₂, x₃)) = 0
POL(U84'(x₁, x₂, x₃)) = x₂
POL(U91'(x₁, x₂)) = x₂
POL(U92'(x₁)) = 0
POL(X(x₁, x₂)) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(c41(x₁)) = x₁
POL(c42(x₁)) = x₁
POL(c43(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(c45(x₁)) = x₁
POL(c46(x₁)) = x₁
POL(c47(x₁)) = x₁
POL(c48(x₁)) = x₁
POL(c49(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c50(x₁)) = x₁
POL(c51(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(c53(x₁)) = x₁
POL(c54(x₁)) = x₁
POL(c55(x₁)) = x₁
POL(c56(x₁)) = x₁
POL(c57(x₁)) = x₁
POL(c58(x₁)) = x₁
POL(c59(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c60(x₁)) = x₁
POL(c61(x₁)) = x₁
POL(c62(x₁)) = x₁
POL(c63(x₁)) = x₁
POL(c64(x₁)) = x₁
POL(c65(x₁)) = x₁
POL(c66(x₁)) = x₁
POL(c67(x₁)) = x₁
POL(c68(x₁)) = x₁
POL(c69(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c70(x₁)) = x₁
POL(c71(x₁)) = x₁
POL(c72(x₁)) = x₁
POL(c73(x₁)) = x₁
POL(c74(x₁)) = x₁
POL(c75(x₁)) = x₁
POL(c76(x₁)) = x₁
POL(c77(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) = [1]

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)

Tuples:

U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
U92'(mark(z0)) → c21(U92'(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(mark(z0)) → c23(U62'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U42'(mark(z0)) → c30(U42'(z0))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
U51'(ok(z0)) → c42(U51'(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
X(z0, mark(z1)) → c47(X(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
U16'(ok(z0)) → c53(U16'(z0))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
S(ok(z0)) → c56(S(z0))
S(mark(z0)) → c57(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U92'(mark(z0)) → c21(U92'(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(mark(z0)) → c23(U62'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U42'(mark(z0)) → c30(U42'(z0))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
U51'(ok(z0)) → c42(U51'(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
X(z0, mark(z1)) → c47(X(z0, z1))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
U16'(ok(z0)) → c53(U16'(z0))
S(ok(z0)) → c56(S(z0))
S(mark(z0)) → c57(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U104', U35', U81', U31', U91', U36', ISNAT, U101', U61', U84', U92', U62', U41', PLUS, U42', U103', ISNATKIND, U82', U11', U51', U15', X, U32', U13', U16', U21', S, U23', U34', U71', U72', U12', U83', U22', U102', U14', U33', TOP

Compound Symbols:

c, c1, 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, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c70, c71, c72, c73, c74, c75, c76, c77, c3

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

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

U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))

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

U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
U92'(mark(z0)) → c21(U92'(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(mark(z0)) → c23(U62'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U42'(mark(z0)) → c30(U42'(z0))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
U51'(ok(z0)) → c42(U51'(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
X(z0, mark(z1)) → c47(X(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
U16'(ok(z0)) → c53(U16'(z0))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
S(ok(z0)) → c56(S(z0))
S(mark(z0)) → c57(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

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

POL(0) = 0
POL(ISNAT(x₁)) = 0
POL(ISNATKIND(x₁)) = 0
POL(PLUS(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(TOP(x₁)) = 0
POL(U101'(x₁, x₂, x₃)) = 0
POL(U102'(x₁, x₂, x₃)) = 0
POL(U103'(x₁, x₂, x₃)) = 0
POL(U104'(x₁, x₂, x₃)) = 0
POL(U11'(x₁, x₂, x₃)) = 0
POL(U12'(x₁, x₂, x₃)) = 0
POL(U13'(x₁, x₂, x₃)) = 0
POL(U14'(x₁, x₂, x₃)) = 0
POL(U15'(x₁, x₂)) = 0
POL(U16'(x₁)) = 0
POL(U21'(x₁, x₂)) = 0
POL(U22'(x₁, x₂)) = 0
POL(U23'(x₁)) = 0
POL(U31'(x₁, x₂, x₃)) = x₃
POL(U32'(x₁, x₂, x₃)) = 0
POL(U33'(x₁, x₂, x₃)) = 0
POL(U34'(x₁, x₂, x₃)) = 0
POL(U35'(x₁, x₂)) = x₁
POL(U36'(x₁)) = 0
POL(U41'(x₁, x₂)) = 0
POL(U42'(x₁)) = 0
POL(U51'(x₁)) = 0
POL(U61'(x₁, x₂)) = 0
POL(U62'(x₁)) = 0
POL(U71'(x₁, x₂)) = x₁
POL(U72'(x₁, x₂)) = 0
POL(U81'(x₁, x₂, x₃)) = 0
POL(U82'(x₁, x₂, x₃)) = 0
POL(U83'(x₁, x₂, x₃)) = 0
POL(U84'(x₁, x₂, x₃)) = 0
POL(U91'(x₁, x₂)) = 0
POL(U92'(x₁)) = 0
POL(X(x₁, x₂)) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(c41(x₁)) = x₁
POL(c42(x₁)) = x₁
POL(c43(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(c45(x₁)) = x₁
POL(c46(x₁)) = x₁
POL(c47(x₁)) = x₁
POL(c48(x₁)) = x₁
POL(c49(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c50(x₁)) = x₁
POL(c51(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(c53(x₁)) = x₁
POL(c54(x₁)) = x₁
POL(c55(x₁)) = x₁
POL(c56(x₁)) = x₁
POL(c57(x₁)) = x₁
POL(c58(x₁)) = x₁
POL(c59(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c60(x₁)) = x₁
POL(c61(x₁)) = x₁
POL(c62(x₁)) = x₁
POL(c63(x₁)) = x₁
POL(c64(x₁)) = x₁
POL(c65(x₁)) = x₁
POL(c66(x₁)) = x₁
POL(c67(x₁)) = x₁
POL(c68(x₁)) = x₁
POL(c69(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c70(x₁)) = x₁
POL(c71(x₁)) = x₁
POL(c72(x₁)) = x₁
POL(c73(x₁)) = x₁
POL(c74(x₁)) = x₁
POL(c75(x₁)) = x₁
POL(c76(x₁)) = x₁
POL(c77(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

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)

Tuples:

U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
U92'(mark(z0)) → c21(U92'(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(mark(z0)) → c23(U62'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U42'(mark(z0)) → c30(U42'(z0))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
U51'(ok(z0)) → c42(U51'(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
X(z0, mark(z1)) → c47(X(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
U16'(ok(z0)) → c53(U16'(z0))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
S(ok(z0)) → c56(S(z0))
S(mark(z0)) → c57(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U92'(mark(z0)) → c21(U92'(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(mark(z0)) → c23(U62'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U42'(mark(z0)) → c30(U42'(z0))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
U51'(ok(z0)) → c42(U51'(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
X(z0, mark(z1)) → c47(X(z0, z1))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
U16'(ok(z0)) → c53(U16'(z0))
S(ok(z0)) → c56(S(z0))
S(mark(z0)) → c57(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U104', U35', U81', U31', U91', U36', ISNAT, U101', U61', U84', U92', U62', U41', PLUS, U42', U103', ISNATKIND, U82', U11', U51', U15', X, U32', U13', U16', U21', S, U23', U34', U71', U72', U12', U83', U22', U102', U14', U33', TOP

Compound Symbols:

c, c1, 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, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c70, c71, c72, c73, c74, c75, c76, c77, c3

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

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

U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U62'(mark(z0)) → c23(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
X(z0, mark(z1)) → c47(X(z0, z1))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
S(mark(z0)) → c57(S(z0))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))

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

U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
U92'(mark(z0)) → c21(U92'(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(mark(z0)) → c23(U62'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U42'(mark(z0)) → c30(U42'(z0))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
U51'(ok(z0)) → c42(U51'(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
X(z0, mark(z1)) → c47(X(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
U16'(ok(z0)) → c53(U16'(z0))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
S(ok(z0)) → c56(S(z0))
S(mark(z0)) → c57(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

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

POL(0) = 0
POL(ISNAT(x₁)) = 0
POL(ISNATKIND(x₁)) = 0
POL(PLUS(x₁, x₂)) = x₁
POL(S(x₁)) = x₁
POL(TOP(x₁)) = 0
POL(U101'(x₁, x₂, x₃)) = 0
POL(U102'(x₁, x₂, x₃)) = 0
POL(U103'(x₁, x₂, x₃)) = 0
POL(U104'(x₁, x₂, x₃)) = x₁
POL(U11'(x₁, x₂, x₃)) = x₁
POL(U12'(x₁, x₂, x₃)) = x₁
POL(U13'(x₁, x₂, x₃)) = x₁
POL(U14'(x₁, x₂, x₃)) = 0
POL(U15'(x₁, x₂)) = 0
POL(U16'(x₁)) = x₁
POL(U21'(x₁, x₂)) = 0
POL(U22'(x₁, x₂)) = x₁
POL(U23'(x₁)) = 0
POL(U31'(x₁, x₂, x₃)) = x₂
POL(U32'(x₁, x₂, x₃)) = 0
POL(U33'(x₁, x₂, x₃)) = x₃
POL(U34'(x₁, x₂, x₃)) = 0
POL(U35'(x₁, x₂)) = 0
POL(U36'(x₁)) = 0
POL(U41'(x₁, x₂)) = x₁
POL(U42'(x₁)) = 0
POL(U51'(x₁)) = x₁
POL(U61'(x₁, x₂)) = 0
POL(U62'(x₁)) = x₁
POL(U71'(x₁, x₂)) = x₁
POL(U72'(x₁, x₂)) = 0
POL(U81'(x₁, x₂, x₃)) = x₁
POL(U82'(x₁, x₂, x₃)) = x₁
POL(U83'(x₁, x₂, x₃)) = x₁
POL(U84'(x₁, x₂, x₃)) = x₁
POL(U91'(x₁, x₂)) = 0
POL(U92'(x₁)) = 0
POL(X(x₁, x₂)) = x₂
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(c41(x₁)) = x₁
POL(c42(x₁)) = x₁
POL(c43(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(c45(x₁)) = x₁
POL(c46(x₁)) = x₁
POL(c47(x₁)) = x₁
POL(c48(x₁)) = x₁
POL(c49(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c50(x₁)) = x₁
POL(c51(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(c53(x₁)) = x₁
POL(c54(x₁)) = x₁
POL(c55(x₁)) = x₁
POL(c56(x₁)) = x₁
POL(c57(x₁)) = x₁
POL(c58(x₁)) = x₁
POL(c59(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c60(x₁)) = x₁
POL(c61(x₁)) = x₁
POL(c62(x₁)) = x₁
POL(c63(x₁)) = x₁
POL(c64(x₁)) = x₁
POL(c65(x₁)) = x₁
POL(c66(x₁)) = x₁
POL(c67(x₁)) = x₁
POL(c68(x₁)) = x₁
POL(c69(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c70(x₁)) = x₁
POL(c71(x₁)) = x₁
POL(c72(x₁)) = x₁
POL(c73(x₁)) = x₁
POL(c74(x₁)) = x₁
POL(c75(x₁)) = x₁
POL(c76(x₁)) = x₁
POL(c77(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

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)

Tuples:

U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
U92'(mark(z0)) → c21(U92'(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(mark(z0)) → c23(U62'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U42'(mark(z0)) → c30(U42'(z0))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
U51'(ok(z0)) → c42(U51'(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
X(z0, mark(z1)) → c47(X(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
U16'(ok(z0)) → c53(U16'(z0))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
S(ok(z0)) → c56(S(z0))
S(mark(z0)) → c57(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U92'(mark(z0)) → c21(U92'(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
U42'(mark(z0)) → c30(U42'(z0))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(ok(z0)) → c42(U51'(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(ok(z0)) → c53(U16'(z0))
S(ok(z0)) → c56(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U62'(mark(z0)) → c23(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
X(z0, mark(z1)) → c47(X(z0, z1))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
S(mark(z0)) → c57(S(z0))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U104', U35', U81', U31', U91', U36', ISNAT, U101', U61', U84', U92', U62', U41', PLUS, U42', U103', ISNATKIND, U82', U11', U51', U15', X, U32', U13', U16', U21', S, U23', U34', U71', U72', U12', U83', U22', U102', U14', U33', TOP

Compound Symbols:

c, c1, 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, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c70, c71, c72, c73, c74, c75, c76, c77, c3

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

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

U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))

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

U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
U92'(mark(z0)) → c21(U92'(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(mark(z0)) → c23(U62'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U42'(mark(z0)) → c30(U42'(z0))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
U51'(ok(z0)) → c42(U51'(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
X(z0, mark(z1)) → c47(X(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
U16'(ok(z0)) → c53(U16'(z0))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
S(ok(z0)) → c56(S(z0))
S(mark(z0)) → c57(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

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

POL(0) = 0
POL(ISNAT(x₁)) = 0
POL(ISNATKIND(x₁)) = 0
POL(PLUS(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(TOP(x₁)) = 0
POL(U101'(x₁, x₂, x₃)) = 0
POL(U102'(x₁, x₂, x₃)) = x₁
POL(U103'(x₁, x₂, x₃)) = 0
POL(U104'(x₁, x₂, x₃)) = 0
POL(U11'(x₁, x₂, x₃)) = 0
POL(U12'(x₁, x₂, x₃)) = 0
POL(U13'(x₁, x₂, x₃)) = 0
POL(U14'(x₁, x₂, x₃)) = 0
POL(U15'(x₁, x₂)) = 0
POL(U16'(x₁)) = 0
POL(U21'(x₁, x₂)) = 0
POL(U22'(x₁, x₂)) = 0
POL(U23'(x₁)) = 0
POL(U31'(x₁, x₂, x₃)) = x₁
POL(U32'(x₁, x₂, x₃)) = 0
POL(U33'(x₁, x₂, x₃)) = x₂ + x₃
POL(U34'(x₁, x₂, x₃)) = 0
POL(U35'(x₁, x₂)) = x₂
POL(U36'(x₁)) = 0
POL(U41'(x₁, x₂)) = 0
POL(U42'(x₁)) = 0
POL(U51'(x₁)) = 0
POL(U61'(x₁, x₂)) = 0
POL(U62'(x₁)) = 0
POL(U71'(x₁, x₂)) = 0
POL(U72'(x₁, x₂)) = 0
POL(U81'(x₁, x₂, x₃)) = 0
POL(U82'(x₁, x₂, x₃)) = 0
POL(U83'(x₁, x₂, x₃)) = 0
POL(U84'(x₁, x₂, x₃)) = x₂
POL(U91'(x₁, x₂)) = x₂
POL(U92'(x₁)) = 0
POL(X(x₁, x₂)) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(c41(x₁)) = x₁
POL(c42(x₁)) = x₁
POL(c43(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(c45(x₁)) = x₁
POL(c46(x₁)) = x₁
POL(c47(x₁)) = x₁
POL(c48(x₁)) = x₁
POL(c49(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c50(x₁)) = x₁
POL(c51(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(c53(x₁)) = x₁
POL(c54(x₁)) = x₁
POL(c55(x₁)) = x₁
POL(c56(x₁)) = x₁
POL(c57(x₁)) = x₁
POL(c58(x₁)) = x₁
POL(c59(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c60(x₁)) = x₁
POL(c61(x₁)) = x₁
POL(c62(x₁)) = x₁
POL(c63(x₁)) = x₁
POL(c64(x₁)) = x₁
POL(c65(x₁)) = x₁
POL(c66(x₁)) = x₁
POL(c67(x₁)) = x₁
POL(c68(x₁)) = x₁
POL(c69(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c70(x₁)) = x₁
POL(c71(x₁)) = x₁
POL(c72(x₁)) = x₁
POL(c73(x₁)) = x₁
POL(c74(x₁)) = x₁
POL(c75(x₁)) = x₁
POL(c76(x₁)) = x₁
POL(c77(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

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)

Tuples:

U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
U92'(mark(z0)) → c21(U92'(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(mark(z0)) → c23(U62'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U42'(mark(z0)) → c30(U42'(z0))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
U51'(ok(z0)) → c42(U51'(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
X(z0, mark(z1)) → c47(X(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
U16'(ok(z0)) → c53(U16'(z0))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
S(ok(z0)) → c56(S(z0))
S(mark(z0)) → c57(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U92'(mark(z0)) → c21(U92'(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
U42'(mark(z0)) → c30(U42'(z0))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(ok(z0)) → c42(U51'(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(ok(z0)) → c53(U16'(z0))
S(ok(z0)) → c56(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U62'(mark(z0)) → c23(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
X(z0, mark(z1)) → c47(X(z0, z1))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
S(mark(z0)) → c57(S(z0))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U104', U35', U81', U31', U91', U36', ISNAT, U101', U61', U84', U92', U62', U41', PLUS, U42', U103', ISNATKIND, U82', U11', U51', U15', X, U32', U13', U16', U21', S, U23', U34', U71', U72', U12', U83', U22', U102', U14', U33', TOP

Compound Symbols:

c, c1, 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, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c70, c71, c72, c73, c74, c75, c76, c77, c3

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

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

U91'(mark(z0), z1) → c10(U91'(z0, z1))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U42'(mark(z0)) → c30(U42'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))

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

U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
U92'(mark(z0)) → c21(U92'(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(mark(z0)) → c23(U62'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U42'(mark(z0)) → c30(U42'(z0))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
U51'(ok(z0)) → c42(U51'(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
X(z0, mark(z1)) → c47(X(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
U16'(ok(z0)) → c53(U16'(z0))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
S(ok(z0)) → c56(S(z0))
S(mark(z0)) → c57(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

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

POL(0) = 0
POL(ISNAT(x₁)) = 0
POL(ISNATKIND(x₁)) = 0
POL(PLUS(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(TOP(x₁)) = 0
POL(U101'(x₁, x₂, x₃)) = x₁
POL(U102'(x₁, x₂, x₃)) = 0
POL(U103'(x₁, x₂, x₃)) = 0
POL(U104'(x₁, x₂, x₃)) = 0
POL(U11'(x₁, x₂, x₃)) = 0
POL(U12'(x₁, x₂, x₃)) = 0
POL(U13'(x₁, x₂, x₃)) = 0
POL(U14'(x₁, x₂, x₃)) = x₁
POL(U15'(x₁, x₂)) = 0
POL(U16'(x₁)) = 0
POL(U21'(x₁, x₂)) = 0
POL(U22'(x₁, x₂)) = 0
POL(U23'(x₁)) = x₁
POL(U31'(x₁, x₂, x₃)) = x₃
POL(U32'(x₁, x₂, x₃)) = x₂
POL(U33'(x₁, x₂, x₃)) = x₂
POL(U34'(x₁, x₂, x₃)) = 0
POL(U35'(x₁, x₂)) = x₂
POL(U36'(x₁)) = x₁
POL(U41'(x₁, x₂)) = 0
POL(U42'(x₁)) = x₁
POL(U51'(x₁)) = 0
POL(U61'(x₁, x₂)) = 0
POL(U62'(x₁)) = 0
POL(U71'(x₁, x₂)) = 0
POL(U72'(x₁, x₂)) = 0
POL(U81'(x₁, x₂, x₃)) = 0
POL(U82'(x₁, x₂, x₃)) = 0
POL(U83'(x₁, x₂, x₃)) = 0
POL(U84'(x₁, x₂, x₃)) = 0
POL(U91'(x₁, x₂)) = x₁
POL(U92'(x₁)) = 0
POL(X(x₁, x₂)) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(c41(x₁)) = x₁
POL(c42(x₁)) = x₁
POL(c43(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(c45(x₁)) = x₁
POL(c46(x₁)) = x₁
POL(c47(x₁)) = x₁
POL(c48(x₁)) = x₁
POL(c49(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c50(x₁)) = x₁
POL(c51(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(c53(x₁)) = x₁
POL(c54(x₁)) = x₁
POL(c55(x₁)) = x₁
POL(c56(x₁)) = x₁
POL(c57(x₁)) = x₁
POL(c58(x₁)) = x₁
POL(c59(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c60(x₁)) = x₁
POL(c61(x₁)) = x₁
POL(c62(x₁)) = x₁
POL(c63(x₁)) = x₁
POL(c64(x₁)) = x₁
POL(c65(x₁)) = x₁
POL(c66(x₁)) = x₁
POL(c67(x₁)) = x₁
POL(c68(x₁)) = x₁
POL(c69(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c70(x₁)) = x₁
POL(c71(x₁)) = x₁
POL(c72(x₁)) = x₁
POL(c73(x₁)) = x₁
POL(c74(x₁)) = x₁
POL(c75(x₁)) = x₁
POL(c76(x₁)) = x₁
POL(c77(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

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)

Tuples:

U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
U92'(mark(z0)) → c21(U92'(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(mark(z0)) → c23(U62'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U42'(mark(z0)) → c30(U42'(z0))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
U51'(ok(z0)) → c42(U51'(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
X(z0, mark(z1)) → c47(X(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
U16'(ok(z0)) → c53(U16'(z0))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
S(ok(z0)) → c56(S(z0))
S(mark(z0)) → c57(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U92'(mark(z0)) → c21(U92'(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(ok(z0)) → c42(U51'(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(ok(z0)) → c53(U16'(z0))
S(ok(z0)) → c56(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U62'(mark(z0)) → c23(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
X(z0, mark(z1)) → c47(X(z0, z1))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
S(mark(z0)) → c57(S(z0))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U42'(mark(z0)) → c30(U42'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U104', U35', U81', U31', U91', U36', ISNAT, U101', U61', U84', U92', U62', U41', PLUS, U42', U103', ISNATKIND, U82', U11', U51', U15', X, U32', U13', U16', U21', S, U23', U34', U71', U72', U12', U83', U22', U102', U14', U33', TOP

Compound Symbols:

c, c1, 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, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c70, c71, c72, c73, c74, c75, c76, c77, c3

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

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

U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))

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

U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
U92'(mark(z0)) → c21(U92'(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(mark(z0)) → c23(U62'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U42'(mark(z0)) → c30(U42'(z0))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
U51'(ok(z0)) → c42(U51'(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
X(z0, mark(z1)) → c47(X(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
U16'(ok(z0)) → c53(U16'(z0))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
S(ok(z0)) → c56(S(z0))
S(mark(z0)) → c57(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

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

POL(0) = 0
POL(ISNAT(x₁)) = 0
POL(ISNATKIND(x₁)) = 0
POL(PLUS(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(TOP(x₁)) = 0
POL(U101'(x₁, x₂, x₃)) = 0
POL(U102'(x₁, x₂, x₃)) = 0
POL(U103'(x₁, x₂, x₃)) = 0
POL(U104'(x₁, x₂, x₃)) = 0
POL(U11'(x₁, x₂, x₃)) = 0
POL(U12'(x₁, x₂, x₃)) = 0
POL(U13'(x₁, x₂, x₃)) = 0
POL(U14'(x₁, x₂, x₃)) = 0
POL(U15'(x₁, x₂)) = 0
POL(U16'(x₁)) = 0
POL(U21'(x₁, x₂)) = x₂
POL(U22'(x₁, x₂)) = 0
POL(U23'(x₁)) = 0
POL(U31'(x₁, x₂, x₃)) = 0
POL(U32'(x₁, x₂, x₃)) = x₁
POL(U33'(x₁, x₂, x₃)) = 0
POL(U34'(x₁, x₂, x₃)) = 0
POL(U35'(x₁, x₂)) = 0
POL(U36'(x₁)) = 0
POL(U41'(x₁, x₂)) = 0
POL(U42'(x₁)) = 0
POL(U51'(x₁)) = 0
POL(U61'(x₁, x₂)) = 0
POL(U62'(x₁)) = 0
POL(U71'(x₁, x₂)) = 0
POL(U72'(x₁, x₂)) = 0
POL(U81'(x₁, x₂, x₃)) = 0
POL(U82'(x₁, x₂, x₃)) = 0
POL(U83'(x₁, x₂, x₃)) = 0
POL(U84'(x₁, x₂, x₃)) = 0
POL(U91'(x₁, x₂)) = 0
POL(U92'(x₁)) = 0
POL(X(x₁, x₂)) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(c41(x₁)) = x₁
POL(c42(x₁)) = x₁
POL(c43(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(c45(x₁)) = x₁
POL(c46(x₁)) = x₁
POL(c47(x₁)) = x₁
POL(c48(x₁)) = x₁
POL(c49(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c50(x₁)) = x₁
POL(c51(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(c53(x₁)) = x₁
POL(c54(x₁)) = x₁
POL(c55(x₁)) = x₁
POL(c56(x₁)) = x₁
POL(c57(x₁)) = x₁
POL(c58(x₁)) = x₁
POL(c59(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c60(x₁)) = x₁
POL(c61(x₁)) = x₁
POL(c62(x₁)) = x₁
POL(c63(x₁)) = x₁
POL(c64(x₁)) = x₁
POL(c65(x₁)) = x₁
POL(c66(x₁)) = x₁
POL(c67(x₁)) = x₁
POL(c68(x₁)) = x₁
POL(c69(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c70(x₁)) = x₁
POL(c71(x₁)) = x₁
POL(c72(x₁)) = x₁
POL(c73(x₁)) = x₁
POL(c74(x₁)) = x₁
POL(c75(x₁)) = x₁
POL(c76(x₁)) = x₁
POL(c77(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

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)

Tuples:

U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
U92'(mark(z0)) → c21(U92'(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(mark(z0)) → c23(U62'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U42'(mark(z0)) → c30(U42'(z0))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
U51'(ok(z0)) → c42(U51'(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
X(z0, mark(z1)) → c47(X(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
U16'(ok(z0)) → c53(U16'(z0))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
S(ok(z0)) → c56(S(z0))
S(mark(z0)) → c57(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U92'(mark(z0)) → c21(U92'(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(ok(z0)) → c42(U51'(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(ok(z0)) → c53(U16'(z0))
S(ok(z0)) → c56(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U62'(mark(z0)) → c23(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
X(z0, mark(z1)) → c47(X(z0, z1))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
S(mark(z0)) → c57(S(z0))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U42'(mark(z0)) → c30(U42'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U104', U35', U81', U31', U91', U36', ISNAT, U101', U61', U84', U92', U62', U41', PLUS, U42', U103', ISNATKIND, U82', U11', U51', U15', X, U32', U13', U16', U21', S, U23', U34', U71', U72', U12', U83', U22', U102', U14', U33', TOP

Compound Symbols:

c, c1, 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, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c70, c71, c72, c73, c74, c75, c76, c77, c3

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

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

U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))

We considered the (Usable) Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)

And the Tuples:

U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
U92'(mark(z0)) → c21(U92'(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(mark(z0)) → c23(U62'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U42'(mark(z0)) → c30(U42'(z0))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
U51'(ok(z0)) → c42(U51'(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
X(z0, mark(z1)) → c47(X(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
U16'(ok(z0)) → c53(U16'(z0))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
S(ok(z0)) → c56(S(z0))
S(mark(z0)) → c57(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

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

POL(0) = 0
POL(ISNAT(x₁)) = 0
POL(ISNATKIND(x₁)) = 0
POL(PLUS(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(TOP(x₁)) = x₁
POL(U101'(x₁, x₂, x₃)) = 0
POL(U102'(x₁, x₂, x₃)) = 0
POL(U103'(x₁, x₂, x₃)) = 0
POL(U104'(x₁, x₂, x₃)) = 0
POL(U11'(x₁, x₂, x₃)) = 0
POL(U12'(x₁, x₂, x₃)) = 0
POL(U13'(x₁, x₂, x₃)) = 0
POL(U14'(x₁, x₂, x₃)) = 0
POL(U15'(x₁, x₂)) = x₁
POL(U16'(x₁)) = 0
POL(U21'(x₁, x₂)) = 0
POL(U22'(x₁, x₂)) = 0
POL(U23'(x₁)) = 0
POL(U31'(x₁, x₂, x₃)) = 0
POL(U32'(x₁, x₂, x₃)) = x₂
POL(U33'(x₁, x₂, x₃)) = 0
POL(U34'(x₁, x₂, x₃)) = 0
POL(U35'(x₁, x₂)) = 0
POL(U36'(x₁)) = 0
POL(U41'(x₁, x₂)) = 0
POL(U42'(x₁)) = 0
POL(U51'(x₁)) = 0
POL(U61'(x₁, x₂)) = 0
POL(U62'(x₁)) = 0
POL(U71'(x₁, x₂)) = 0
POL(U72'(x₁, x₂)) = 0
POL(U81'(x₁, x₂, x₃)) = 0
POL(U82'(x₁, x₂, x₃)) = 0
POL(U83'(x₁, x₂, x₃)) = 0
POL(U84'(x₁, x₂, x₃)) = 0
POL(U91'(x₁, x₂)) = 0
POL(U92'(x₁)) = 0
POL(X(x₁, x₂)) = x₁
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(c41(x₁)) = x₁
POL(c42(x₁)) = x₁
POL(c43(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(c45(x₁)) = x₁
POL(c46(x₁)) = x₁
POL(c47(x₁)) = x₁
POL(c48(x₁)) = x₁
POL(c49(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c50(x₁)) = x₁
POL(c51(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(c53(x₁)) = x₁
POL(c54(x₁)) = x₁
POL(c55(x₁)) = x₁
POL(c56(x₁)) = x₁
POL(c57(x₁)) = x₁
POL(c58(x₁)) = x₁
POL(c59(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c60(x₁)) = x₁
POL(c61(x₁)) = x₁
POL(c62(x₁)) = x₁
POL(c63(x₁)) = x₁
POL(c64(x₁)) = x₁
POL(c65(x₁)) = x₁
POL(c66(x₁)) = x₁
POL(c67(x₁)) = x₁
POL(c68(x₁)) = x₁
POL(c69(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c70(x₁)) = x₁
POL(c71(x₁)) = x₁
POL(c72(x₁)) = x₁
POL(c73(x₁)) = x₁
POL(c74(x₁)) = x₁
POL(c75(x₁)) = x₁
POL(c76(x₁)) = x₁
POL(c77(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

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)

Tuples:

U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
U92'(mark(z0)) → c21(U92'(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(mark(z0)) → c23(U62'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U42'(mark(z0)) → c30(U42'(z0))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
U51'(ok(z0)) → c42(U51'(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
X(z0, mark(z1)) → c47(X(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
U16'(ok(z0)) → c53(U16'(z0))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
S(ok(z0)) → c56(S(z0))
S(mark(z0)) → c57(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U92'(mark(z0)) → c21(U92'(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(ok(z0)) → c42(U51'(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(ok(z0)) → c53(U16'(z0))
S(ok(z0)) → c56(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U62'(mark(z0)) → c23(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
X(z0, mark(z1)) → c47(X(z0, z1))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
S(mark(z0)) → c57(S(z0))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U42'(mark(z0)) → c30(U42'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U104', U35', U81', U31', U91', U36', ISNAT, U101', U61', U84', U92', U62', U41', PLUS, U42', U103', ISNATKIND, U82', U11', U51', U15', X, U32', U13', U16', U21', S, U23', U34', U71', U72', U12', U83', U22', U102', U14', U33', TOP

Compound Symbols:

c, c1, 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, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c70, c71, c72, c73, c74, c75, c76, c77, c3

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

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

U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))

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

U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
U92'(mark(z0)) → c21(U92'(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(mark(z0)) → c23(U62'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U42'(mark(z0)) → c30(U42'(z0))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
U51'(ok(z0)) → c42(U51'(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
X(z0, mark(z1)) → c47(X(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
U16'(ok(z0)) → c53(U16'(z0))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
S(ok(z0)) → c56(S(z0))
S(mark(z0)) → c57(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

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

POL(0) = 0
POL(ISNAT(x₁)) = 0
POL(ISNATKIND(x₁)) = x₁
POL(PLUS(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(TOP(x₁)) = 0
POL(U101'(x₁, x₂, x₃)) = x₃
POL(U102'(x₁, x₂, x₃)) = 0
POL(U103'(x₁, x₂, x₃)) = x₃
POL(U104'(x₁, x₂, x₃)) = x₂
POL(U11'(x₁, x₂, x₃)) = 0
POL(U12'(x₁, x₂, x₃)) = 0
POL(U13'(x₁, x₂, x₃)) = 0
POL(U14'(x₁, x₂, x₃)) = x₁
POL(U15'(x₁, x₂)) = x₁ + x₂
POL(U16'(x₁)) = 0
POL(U21'(x₁, x₂)) = 0
POL(U22'(x₁, x₂)) = 0
POL(U23'(x₁)) = 0
POL(U31'(x₁, x₂, x₃)) = 0
POL(U32'(x₁, x₂, x₃)) = x₂
POL(U33'(x₁, x₂, x₃)) = 0
POL(U34'(x₁, x₂, x₃)) = x₃
POL(U35'(x₁, x₂)) = 0
POL(U36'(x₁)) = 0
POL(U41'(x₁, x₂)) = x₁ + x₂
POL(U42'(x₁)) = 0
POL(U51'(x₁)) = 0
POL(U61'(x₁, x₂)) = x₁
POL(U62'(x₁)) = 0
POL(U71'(x₁, x₂)) = x₂
POL(U72'(x₁, x₂)) = 0
POL(U81'(x₁, x₂, x₃)) = x₃
POL(U82'(x₁, x₂, x₃)) = 0
POL(U83'(x₁, x₂, x₃)) = x₁ + x₂
POL(U84'(x₁, x₂, x₃)) = 0
POL(U91'(x₁, x₂)) = x₂
POL(U92'(x₁)) = 0
POL(X(x₁, x₂)) = x₁ + x₂
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(c41(x₁)) = x₁
POL(c42(x₁)) = x₁
POL(c43(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(c45(x₁)) = x₁
POL(c46(x₁)) = x₁
POL(c47(x₁)) = x₁
POL(c48(x₁)) = x₁
POL(c49(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c50(x₁)) = x₁
POL(c51(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(c53(x₁)) = x₁
POL(c54(x₁)) = x₁
POL(c55(x₁)) = x₁
POL(c56(x₁)) = x₁
POL(c57(x₁)) = x₁
POL(c58(x₁)) = x₁
POL(c59(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c60(x₁)) = x₁
POL(c61(x₁)) = x₁
POL(c62(x₁)) = x₁
POL(c63(x₁)) = x₁
POL(c64(x₁)) = x₁
POL(c65(x₁)) = x₁
POL(c66(x₁)) = x₁
POL(c67(x₁)) = x₁
POL(c68(x₁)) = x₁
POL(c69(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c70(x₁)) = x₁
POL(c71(x₁)) = x₁
POL(c72(x₁)) = x₁
POL(c73(x₁)) = x₁
POL(c74(x₁)) = x₁
POL(c75(x₁)) = x₁
POL(c76(x₁)) = x₁
POL(c77(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

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)

Tuples:

U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
U92'(mark(z0)) → c21(U92'(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(mark(z0)) → c23(U62'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U42'(mark(z0)) → c30(U42'(z0))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
U51'(ok(z0)) → c42(U51'(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
X(z0, mark(z1)) → c47(X(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
U16'(ok(z0)) → c53(U16'(z0))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
S(ok(z0)) → c56(S(z0))
S(mark(z0)) → c57(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U35'(mark(z0), z1) → c5(U35'(z0, z1))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U92'(mark(z0)) → c21(U92'(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(ok(z0)) → c42(U51'(z0))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(ok(z0)) → c53(U16'(z0))
S(ok(z0)) → c56(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U62'(mark(z0)) → c23(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
X(z0, mark(z1)) → c47(X(z0, z1))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
S(mark(z0)) → c57(S(z0))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U42'(mark(z0)) → c30(U42'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U104', U35', U81', U31', U91', U36', ISNAT, U101', U61', U84', U92', U62', U41', PLUS, U42', U103', ISNATKIND, U82', U11', U51', U15', X, U32', U13', U16', U21', S, U23', U34', U71', U72', U12', U83', U22', U102', U14', U33', TOP

Compound Symbols:

c, c1, 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, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c70, c71, c72, c73, c74, c75, c76, c77, c3

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

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

U35'(mark(z0), z1) → c5(U35'(z0, z1))

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

U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
U92'(mark(z0)) → c21(U92'(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(mark(z0)) → c23(U62'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U42'(mark(z0)) → c30(U42'(z0))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
U51'(ok(z0)) → c42(U51'(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
X(z0, mark(z1)) → c47(X(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
U16'(ok(z0)) → c53(U16'(z0))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
S(ok(z0)) → c56(S(z0))
S(mark(z0)) → c57(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

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

POL(0) = 0
POL(ISNAT(x₁)) = 0
POL(ISNATKIND(x₁)) = 0
POL(PLUS(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(TOP(x₁)) = 0
POL(U101'(x₁, x₂, x₃)) = 0
POL(U102'(x₁, x₂, x₃)) = 0
POL(U103'(x₁, x₂, x₃)) = 0
POL(U104'(x₁, x₂, x₃)) = 0
POL(U11'(x₁, x₂, x₃)) = 0
POL(U12'(x₁, x₂, x₃)) = 0
POL(U13'(x₁, x₂, x₃)) = 0
POL(U14'(x₁, x₂, x₃)) = 0
POL(U15'(x₁, x₂)) = 0
POL(U16'(x₁)) = 0
POL(U21'(x₁, x₂)) = 0
POL(U22'(x₁, x₂)) = 0
POL(U23'(x₁)) = 0
POL(U31'(x₁, x₂, x₃)) = 0
POL(U32'(x₁, x₂, x₃)) = x₂
POL(U33'(x₁, x₂, x₃)) = 0
POL(U34'(x₁, x₂, x₃)) = 0
POL(U35'(x₁, x₂)) = x₁
POL(U36'(x₁)) = 0
POL(U41'(x₁, x₂)) = 0
POL(U42'(x₁)) = 0
POL(U51'(x₁)) = 0
POL(U61'(x₁, x₂)) = 0
POL(U62'(x₁)) = 0
POL(U71'(x₁, x₂)) = 0
POL(U72'(x₁, x₂)) = 0
POL(U81'(x₁, x₂, x₃)) = x₂
POL(U82'(x₁, x₂, x₃)) = 0
POL(U83'(x₁, x₂, x₃)) = 0
POL(U84'(x₁, x₂, x₃)) = 0
POL(U91'(x₁, x₂)) = 0
POL(U92'(x₁)) = 0
POL(X(x₁, x₂)) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(c41(x₁)) = x₁
POL(c42(x₁)) = x₁
POL(c43(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(c45(x₁)) = x₁
POL(c46(x₁)) = x₁
POL(c47(x₁)) = x₁
POL(c48(x₁)) = x₁
POL(c49(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c50(x₁)) = x₁
POL(c51(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(c53(x₁)) = x₁
POL(c54(x₁)) = x₁
POL(c55(x₁)) = x₁
POL(c56(x₁)) = x₁
POL(c57(x₁)) = x₁
POL(c58(x₁)) = x₁
POL(c59(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c60(x₁)) = x₁
POL(c61(x₁)) = x₁
POL(c62(x₁)) = x₁
POL(c63(x₁)) = x₁
POL(c64(x₁)) = x₁
POL(c65(x₁)) = x₁
POL(c66(x₁)) = x₁
POL(c67(x₁)) = x₁
POL(c68(x₁)) = x₁
POL(c69(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c70(x₁)) = x₁
POL(c71(x₁)) = x₁
POL(c72(x₁)) = x₁
POL(c73(x₁)) = x₁
POL(c74(x₁)) = x₁
POL(c75(x₁)) = x₁
POL(c76(x₁)) = x₁
POL(c77(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

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)

Tuples:

U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
U92'(mark(z0)) → c21(U92'(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(mark(z0)) → c23(U62'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U42'(mark(z0)) → c30(U42'(z0))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
U51'(ok(z0)) → c42(U51'(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
X(z0, mark(z1)) → c47(X(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
U16'(ok(z0)) → c53(U16'(z0))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
S(ok(z0)) → c56(S(z0))
S(mark(z0)) → c57(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

ISNAT(ok(z0)) → c14(ISNAT(z0))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U92'(mark(z0)) → c21(U92'(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(ok(z0)) → c42(U51'(z0))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(ok(z0)) → c53(U16'(z0))
S(ok(z0)) → c56(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U62'(mark(z0)) → c23(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
X(z0, mark(z1)) → c47(X(z0, z1))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
S(mark(z0)) → c57(S(z0))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U42'(mark(z0)) → c30(U42'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U35'(mark(z0), z1) → c5(U35'(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U104', U35', U81', U31', U91', U36', ISNAT, U101', U61', U84', U92', U62', U41', PLUS, U42', U103', ISNATKIND, U82', U11', U51', U15', X, U32', U13', U16', U21', S, U23', U34', U71', U72', U12', U83', U22', U102', U14', U33', TOP

Compound Symbols:

c, c1, 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, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c70, c71, c72, c73, c74, c75, c76, c77, c3

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

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

U72'(mark(z0), z1) → c64(U72'(z0, z1))

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

U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
U92'(mark(z0)) → c21(U92'(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(mark(z0)) → c23(U62'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U42'(mark(z0)) → c30(U42'(z0))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
U51'(ok(z0)) → c42(U51'(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
X(z0, mark(z1)) → c47(X(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
U16'(ok(z0)) → c53(U16'(z0))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
S(ok(z0)) → c56(S(z0))
S(mark(z0)) → c57(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

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

POL(0) = [1]
POL(ISNAT(x₁)) = 0
POL(ISNATKIND(x₁)) = 0
POL(PLUS(x₁, x₂)) = x₁
POL(S(x₁)) = 0
POL(TOP(x₁)) = 0
POL(U101'(x₁, x₂, x₃)) = 0
POL(U102'(x₁, x₂, x₃)) = 0
POL(U103'(x₁, x₂, x₃)) = x₂
POL(U104'(x₁, x₂, x₃)) = 0
POL(U11'(x₁, x₂, x₃)) = 0
POL(U12'(x₁, x₂, x₃)) = 0
POL(U13'(x₁, x₂, x₃)) = 0
POL(U14'(x₁, x₂, x₃)) = 0
POL(U15'(x₁, x₂)) = 0
POL(U16'(x₁)) = 0
POL(U21'(x₁, x₂)) = 0
POL(U22'(x₁, x₂)) = 0
POL(U23'(x₁)) = 0
POL(U31'(x₁, x₂, x₃)) = 0
POL(U32'(x₁, x₂, x₃)) = 0
POL(U33'(x₁, x₂, x₃)) = 0
POL(U34'(x₁, x₂, x₃)) = 0
POL(U35'(x₁, x₂)) = 0
POL(U36'(x₁)) = 0
POL(U41'(x₁, x₂)) = 0
POL(U42'(x₁)) = 0
POL(U51'(x₁)) = 0
POL(U61'(x₁, x₂)) = 0
POL(U62'(x₁)) = 0
POL(U71'(x₁, x₂)) = x₁
POL(U72'(x₁, x₂)) = x₁
POL(U81'(x₁, x₂, x₃)) = 0
POL(U82'(x₁, x₂, x₃)) = 0
POL(U83'(x₁, x₂, x₃)) = x₂
POL(U84'(x₁, x₂, x₃)) = 0
POL(U91'(x₁, x₂)) = 0
POL(U92'(x₁)) = 0
POL(X(x₁, x₂)) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(c41(x₁)) = x₁
POL(c42(x₁)) = x₁
POL(c43(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(c45(x₁)) = x₁
POL(c46(x₁)) = x₁
POL(c47(x₁)) = x₁
POL(c48(x₁)) = x₁
POL(c49(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c50(x₁)) = x₁
POL(c51(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(c53(x₁)) = x₁
POL(c54(x₁)) = x₁
POL(c55(x₁)) = x₁
POL(c56(x₁)) = x₁
POL(c57(x₁)) = x₁
POL(c58(x₁)) = x₁
POL(c59(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c60(x₁)) = x₁
POL(c61(x₁)) = x₁
POL(c62(x₁)) = x₁
POL(c63(x₁)) = x₁
POL(c64(x₁)) = x₁
POL(c65(x₁)) = x₁
POL(c66(x₁)) = x₁
POL(c67(x₁)) = x₁
POL(c68(x₁)) = x₁
POL(c69(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c70(x₁)) = x₁
POL(c71(x₁)) = x₁
POL(c72(x₁)) = x₁
POL(c73(x₁)) = x₁
POL(c74(x₁)) = x₁
POL(c75(x₁)) = x₁
POL(c76(x₁)) = x₁
POL(c77(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) = [1]

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)

Tuples:

U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
U92'(mark(z0)) → c21(U92'(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(mark(z0)) → c23(U62'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U42'(mark(z0)) → c30(U42'(z0))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
U51'(ok(z0)) → c42(U51'(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
X(z0, mark(z1)) → c47(X(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
U16'(ok(z0)) → c53(U16'(z0))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
S(ok(z0)) → c56(S(z0))
S(mark(z0)) → c57(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

ISNAT(ok(z0)) → c14(ISNAT(z0))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U92'(mark(z0)) → c21(U92'(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(ok(z0)) → c42(U51'(z0))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(ok(z0)) → c53(U16'(z0))
S(ok(z0)) → c56(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U62'(mark(z0)) → c23(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
X(z0, mark(z1)) → c47(X(z0, z1))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
S(mark(z0)) → c57(S(z0))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U42'(mark(z0)) → c30(U42'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U72'(mark(z0), z1) → c64(U72'(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U104', U35', U81', U31', U91', U36', ISNAT, U101', U61', U84', U92', U62', U41', PLUS, U42', U103', ISNATKIND, U82', U11', U51', U15', X, U32', U13', U16', U21', S, U23', U34', U71', U72', U12', U83', U22', U102', U14', U33', TOP

Compound Symbols:

c, c1, 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, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c70, c71, c72, c73, c74, c75, c76, c77, c3

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

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

U92'(mark(z0)) → c21(U92'(z0))

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

U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
U92'(mark(z0)) → c21(U92'(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(mark(z0)) → c23(U62'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U42'(mark(z0)) → c30(U42'(z0))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
U51'(ok(z0)) → c42(U51'(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
X(z0, mark(z1)) → c47(X(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
U16'(ok(z0)) → c53(U16'(z0))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
S(ok(z0)) → c56(S(z0))
S(mark(z0)) → c57(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

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

POL(0) = [1]
POL(ISNAT(x₁)) = 0
POL(ISNATKIND(x₁)) = 0
POL(PLUS(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(TOP(x₁)) = 0
POL(U101'(x₁, x₂, x₃)) = 0
POL(U102'(x₁, x₂, x₃)) = 0
POL(U103'(x₁, x₂, x₃)) = x₃
POL(U104'(x₁, x₂, x₃)) = 0
POL(U11'(x₁, x₂, x₃)) = 0
POL(U12'(x₁, x₂, x₃)) = 0
POL(U13'(x₁, x₂, x₃)) = 0
POL(U14'(x₁, x₂, x₃)) = 0
POL(U15'(x₁, x₂)) = 0
POL(U16'(x₁)) = 0
POL(U21'(x₁, x₂)) = 0
POL(U22'(x₁, x₂)) = 0
POL(U23'(x₁)) = 0
POL(U31'(x₁, x₂, x₃)) = 0
POL(U32'(x₁, x₂, x₃)) = 0
POL(U33'(x₁, x₂, x₃)) = 0
POL(U34'(x₁, x₂, x₃)) = 0
POL(U35'(x₁, x₂)) = 0
POL(U36'(x₁)) = 0
POL(U41'(x₁, x₂)) = 0
POL(U42'(x₁)) = 0
POL(U51'(x₁)) = 0
POL(U61'(x₁, x₂)) = 0
POL(U62'(x₁)) = 0
POL(U71'(x₁, x₂)) = 0
POL(U72'(x₁, x₂)) = 0
POL(U81'(x₁, x₂, x₃)) = x₂
POL(U82'(x₁, x₂, x₃)) = 0
POL(U83'(x₁, x₂, x₃)) = 0
POL(U84'(x₁, x₂, x₃)) = 0
POL(U91'(x₁, x₂)) = 0
POL(U92'(x₁)) = x₁
POL(X(x₁, x₂)) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(c41(x₁)) = x₁
POL(c42(x₁)) = x₁
POL(c43(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(c45(x₁)) = x₁
POL(c46(x₁)) = x₁
POL(c47(x₁)) = x₁
POL(c48(x₁)) = x₁
POL(c49(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c50(x₁)) = x₁
POL(c51(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(c53(x₁)) = x₁
POL(c54(x₁)) = x₁
POL(c55(x₁)) = x₁
POL(c56(x₁)) = x₁
POL(c57(x₁)) = x₁
POL(c58(x₁)) = x₁
POL(c59(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c60(x₁)) = x₁
POL(c61(x₁)) = x₁
POL(c62(x₁)) = x₁
POL(c63(x₁)) = x₁
POL(c64(x₁)) = x₁
POL(c65(x₁)) = x₁
POL(c66(x₁)) = x₁
POL(c67(x₁)) = x₁
POL(c68(x₁)) = x₁
POL(c69(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c70(x₁)) = x₁
POL(c71(x₁)) = x₁
POL(c72(x₁)) = x₁
POL(c73(x₁)) = x₁
POL(c74(x₁)) = x₁
POL(c75(x₁)) = x₁
POL(c76(x₁)) = x₁
POL(c77(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

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)

Tuples:

U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
U92'(mark(z0)) → c21(U92'(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(mark(z0)) → c23(U62'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U42'(mark(z0)) → c30(U42'(z0))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
U51'(ok(z0)) → c42(U51'(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
X(z0, mark(z1)) → c47(X(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
U16'(ok(z0)) → c53(U16'(z0))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
S(ok(z0)) → c56(S(z0))
S(mark(z0)) → c57(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

ISNAT(ok(z0)) → c14(ISNAT(z0))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U92'(ok(z0)) → c22(U92'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(ok(z0)) → c42(U51'(z0))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(ok(z0)) → c53(U16'(z0))
S(ok(z0)) → c56(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U62'(mark(z0)) → c23(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
X(z0, mark(z1)) → c47(X(z0, z1))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
S(mark(z0)) → c57(S(z0))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U42'(mark(z0)) → c30(U42'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U92'(mark(z0)) → c21(U92'(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U104', U35', U81', U31', U91', U36', ISNAT, U101', U61', U84', U92', U62', U41', PLUS, U42', U103', ISNATKIND, U82', U11', U51', U15', X, U32', U13', U16', U21', S, U23', U34', U71', U72', U12', U83', U22', U102', U14', U33', TOP

Compound Symbols:

c, c1, 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, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c70, c71, c72, c73, c74, c75, c76, c77, c3

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

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

U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))

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

U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
U92'(mark(z0)) → c21(U92'(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(mark(z0)) → c23(U62'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U42'(mark(z0)) → c30(U42'(z0))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
U51'(ok(z0)) → c42(U51'(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
X(z0, mark(z1)) → c47(X(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
U16'(ok(z0)) → c53(U16'(z0))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
S(ok(z0)) → c56(S(z0))
S(mark(z0)) → c57(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

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

POL(0) = 0
POL(ISNAT(x₁)) = 0
POL(ISNATKIND(x₁)) = 0
POL(PLUS(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(TOP(x₁)) = 0
POL(U101'(x₁, x₂, x₃)) = 0
POL(U102'(x₁, x₂, x₃)) = 0
POL(U103'(x₁, x₂, x₃)) = x₁
POL(U104'(x₁, x₂, x₃)) = x₂
POL(U11'(x₁, x₂, x₃)) = 0
POL(U12'(x₁, x₂, x₃)) = 0
POL(U13'(x₁, x₂, x₃)) = 0
POL(U14'(x₁, x₂, x₃)) = 0
POL(U15'(x₁, x₂)) = 0
POL(U16'(x₁)) = 0
POL(U21'(x₁, x₂)) = 0
POL(U22'(x₁, x₂)) = 0
POL(U23'(x₁)) = 0
POL(U31'(x₁, x₂, x₃)) = 0
POL(U32'(x₁, x₂, x₃)) = 0
POL(U33'(x₁, x₂, x₃)) = 0
POL(U34'(x₁, x₂, x₃)) = 0
POL(U35'(x₁, x₂)) = 0
POL(U36'(x₁)) = 0
POL(U41'(x₁, x₂)) = 0
POL(U42'(x₁)) = 0
POL(U51'(x₁)) = 0
POL(U61'(x₁, x₂)) = 0
POL(U62'(x₁)) = 0
POL(U71'(x₁, x₂)) = 0
POL(U72'(x₁, x₂)) = 0
POL(U81'(x₁, x₂, x₃)) = 0
POL(U82'(x₁, x₂, x₃)) = 0
POL(U83'(x₁, x₂, x₃)) = x₃
POL(U84'(x₁, x₂, x₃)) = 0
POL(U91'(x₁, x₂)) = 0
POL(U92'(x₁)) = 0
POL(X(x₁, x₂)) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(c41(x₁)) = x₁
POL(c42(x₁)) = x₁
POL(c43(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(c45(x₁)) = x₁
POL(c46(x₁)) = x₁
POL(c47(x₁)) = x₁
POL(c48(x₁)) = x₁
POL(c49(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c50(x₁)) = x₁
POL(c51(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(c53(x₁)) = x₁
POL(c54(x₁)) = x₁
POL(c55(x₁)) = x₁
POL(c56(x₁)) = x₁
POL(c57(x₁)) = x₁
POL(c58(x₁)) = x₁
POL(c59(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c60(x₁)) = x₁
POL(c61(x₁)) = x₁
POL(c62(x₁)) = x₁
POL(c63(x₁)) = x₁
POL(c64(x₁)) = x₁
POL(c65(x₁)) = x₁
POL(c66(x₁)) = x₁
POL(c67(x₁)) = x₁
POL(c68(x₁)) = x₁
POL(c69(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c70(x₁)) = x₁
POL(c71(x₁)) = x₁
POL(c72(x₁)) = x₁
POL(c73(x₁)) = x₁
POL(c74(x₁)) = x₁
POL(c75(x₁)) = x₁
POL(c76(x₁)) = x₁
POL(c77(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:

U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
U92'(mark(z0)) → c21(U92'(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(mark(z0)) → c23(U62'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U42'(mark(z0)) → c30(U42'(z0))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
U51'(ok(z0)) → c42(U51'(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
X(z0, mark(z1)) → c47(X(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
U16'(ok(z0)) → c53(U16'(z0))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
S(ok(z0)) → c56(S(z0))
S(mark(z0)) → c57(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

ISNAT(ok(z0)) → c14(ISNAT(z0))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U92'(ok(z0)) → c22(U92'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U42'(ok(z0)) → c31(U42'(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(ok(z0)) → c42(U51'(z0))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(ok(z0)) → c53(U16'(z0))
S(ok(z0)) → c56(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U62'(mark(z0)) → c23(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
X(z0, mark(z1)) → c47(X(z0, z1))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
S(mark(z0)) → c57(S(z0))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U42'(mark(z0)) → c30(U42'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U92'(mark(z0)) → c21(U92'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U104', U35', U81', U31', U91', U36', ISNAT, U101', U61', U84', U92', U62', U41', PLUS, U42', U103', ISNATKIND, U82', U11', U51', U15', X, U32', U13', U16', U21', S, U23', U34', U71', U72', U12', U83', U22', U102', U14', U33', TOP

Compound Symbols:

c, c1, 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, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c70, c71, c72, c73, c74, c75, c76, c77, c3

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

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

U61'(mark(z0), z1) → c18(U61'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))

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

U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
U92'(mark(z0)) → c21(U92'(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(mark(z0)) → c23(U62'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U42'(mark(z0)) → c30(U42'(z0))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
U51'(ok(z0)) → c42(U51'(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
X(z0, mark(z1)) → c47(X(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
U16'(ok(z0)) → c53(U16'(z0))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
S(ok(z0)) → c56(S(z0))
S(mark(z0)) → c57(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

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

POL(0) = 0
POL(ISNAT(x₁)) = 0
POL(ISNATKIND(x₁)) = 0
POL(PLUS(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(TOP(x₁)) = 0
POL(U101'(x₁, x₂, x₃)) = 0
POL(U102'(x₁, x₂, x₃)) = 0
POL(U103'(x₁, x₂, x₃)) = x₂
POL(U104'(x₁, x₂, x₃)) = 0
POL(U11'(x₁, x₂, x₃)) = 0
POL(U12'(x₁, x₂, x₃)) = 0
POL(U13'(x₁, x₂, x₃)) = 0
POL(U14'(x₁, x₂, x₃)) = 0
POL(U15'(x₁, x₂)) = 0
POL(U16'(x₁)) = 0
POL(U21'(x₁, x₂)) = 0
POL(U22'(x₁, x₂)) = 0
POL(U23'(x₁)) = 0
POL(U31'(x₁, x₂, x₃)) = x₂
POL(U32'(x₁, x₂, x₃)) = 0
POL(U33'(x₁, x₂, x₃)) = 0
POL(U34'(x₁, x₂, x₃)) = 0
POL(U35'(x₁, x₂)) = 0
POL(U36'(x₁)) = 0
POL(U41'(x₁, x₂)) = 0
POL(U42'(x₁)) = 0
POL(U51'(x₁)) = 0
POL(U61'(x₁, x₂)) = x₁
POL(U62'(x₁)) = 0
POL(U71'(x₁, x₂)) = 0
POL(U72'(x₁, x₂)) = x₁
POL(U81'(x₁, x₂, x₃)) = 0
POL(U82'(x₁, x₂, x₃)) = 0
POL(U83'(x₁, x₂, x₃)) = 0
POL(U84'(x₁, x₂, x₃)) = 0
POL(U91'(x₁, x₂)) = 0
POL(U92'(x₁)) = 0
POL(X(x₁, x₂)) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(c41(x₁)) = x₁
POL(c42(x₁)) = x₁
POL(c43(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(c45(x₁)) = x₁
POL(c46(x₁)) = x₁
POL(c47(x₁)) = x₁
POL(c48(x₁)) = x₁
POL(c49(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c50(x₁)) = x₁
POL(c51(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(c53(x₁)) = x₁
POL(c54(x₁)) = x₁
POL(c55(x₁)) = x₁
POL(c56(x₁)) = x₁
POL(c57(x₁)) = x₁
POL(c58(x₁)) = x₁
POL(c59(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c60(x₁)) = x₁
POL(c61(x₁)) = x₁
POL(c62(x₁)) = x₁
POL(c63(x₁)) = x₁
POL(c64(x₁)) = x₁
POL(c65(x₁)) = x₁
POL(c66(x₁)) = x₁
POL(c67(x₁)) = x₁
POL(c68(x₁)) = x₁
POL(c69(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c70(x₁)) = x₁
POL(c71(x₁)) = x₁
POL(c72(x₁)) = x₁
POL(c73(x₁)) = x₁
POL(c74(x₁)) = x₁
POL(c75(x₁)) = x₁
POL(c76(x₁)) = x₁
POL(c77(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

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)

Tuples:

U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
U92'(mark(z0)) → c21(U92'(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(mark(z0)) → c23(U62'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U42'(mark(z0)) → c30(U42'(z0))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
U51'(ok(z0)) → c42(U51'(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
X(z0, mark(z1)) → c47(X(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
U16'(ok(z0)) → c53(U16'(z0))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
S(ok(z0)) → c56(S(z0))
S(mark(z0)) → c57(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

ISNAT(ok(z0)) → c14(ISNAT(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U42'(ok(z0)) → c31(U42'(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(ok(z0)) → c42(U51'(z0))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(ok(z0)) → c53(U16'(z0))
S(ok(z0)) → c56(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U62'(mark(z0)) → c23(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
X(z0, mark(z1)) → c47(X(z0, z1))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
S(mark(z0)) → c57(S(z0))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U42'(mark(z0)) → c30(U42'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U92'(mark(z0)) → c21(U92'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U104', U35', U81', U31', U91', U36', ISNAT, U101', U61', U84', U92', U62', U41', PLUS, U42', U103', ISNATKIND, U82', U11', U51', U15', X, U32', U13', U16', U21', S, U23', U34', U71', U72', U12', U83', U22', U102', U14', U33', TOP

Compound Symbols:

c, c1, 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, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c70, c71, c72, c73, c74, c75, c76, c77, c3

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

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

ISNAT(ok(z0)) → c14(ISNAT(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(ok(z0)) → c42(U51'(z0))
S(ok(z0)) → c56(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))

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

U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
U92'(mark(z0)) → c21(U92'(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(mark(z0)) → c23(U62'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U42'(mark(z0)) → c30(U42'(z0))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
U51'(ok(z0)) → c42(U51'(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
X(z0, mark(z1)) → c47(X(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
U16'(ok(z0)) → c53(U16'(z0))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
S(ok(z0)) → c56(S(z0))
S(mark(z0)) → c57(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

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

POL(0) = 0
POL(ISNAT(x₁)) = x₁
POL(ISNATKIND(x₁)) = 0
POL(PLUS(x₁, x₂)) = x₂
POL(S(x₁)) = x₁
POL(TOP(x₁)) = 0
POL(U101'(x₁, x₂, x₃)) = 0
POL(U102'(x₁, x₂, x₃)) = x₃
POL(U103'(x₁, x₂, x₃)) = 0
POL(U104'(x₁, x₂, x₃)) = x₂ + x₃
POL(U11'(x₁, x₂, x₃)) = x₃
POL(U12'(x₁, x₂, x₃)) = 0
POL(U13'(x₁, x₂, x₃)) = 0
POL(U14'(x₁, x₂, x₃)) = x₃
POL(U15'(x₁, x₂)) = 0
POL(U16'(x₁)) = 0
POL(U21'(x₁, x₂)) = 0
POL(U22'(x₁, x₂)) = x₁
POL(U23'(x₁)) = x₁
POL(U31'(x₁, x₂, x₃)) = 0
POL(U32'(x₁, x₂, x₃)) = 0
POL(U33'(x₁, x₂, x₃)) = 0
POL(U34'(x₁, x₂, x₃)) = 0
POL(U35'(x₁, x₂)) = 0
POL(U36'(x₁)) = 0
POL(U41'(x₁, x₂)) = 0
POL(U42'(x₁)) = 0
POL(U51'(x₁)) = x₁
POL(U61'(x₁, x₂)) = x₁ + x₂
POL(U62'(x₁)) = x₁
POL(U71'(x₁, x₂)) = 0
POL(U72'(x₁, x₂)) = 0
POL(U81'(x₁, x₂, x₃)) = x₂
POL(U82'(x₁, x₂, x₃)) = x₁
POL(U83'(x₁, x₂, x₃)) = 0
POL(U84'(x₁, x₂, x₃)) = 0
POL(U91'(x₁, x₂)) = 0
POL(U92'(x₁)) = x₁
POL(X(x₁, x₂)) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(c41(x₁)) = x₁
POL(c42(x₁)) = x₁
POL(c43(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(c45(x₁)) = x₁
POL(c46(x₁)) = x₁
POL(c47(x₁)) = x₁
POL(c48(x₁)) = x₁
POL(c49(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c50(x₁)) = x₁
POL(c51(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(c53(x₁)) = x₁
POL(c54(x₁)) = x₁
POL(c55(x₁)) = x₁
POL(c56(x₁)) = x₁
POL(c57(x₁)) = x₁
POL(c58(x₁)) = x₁
POL(c59(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c60(x₁)) = x₁
POL(c61(x₁)) = x₁
POL(c62(x₁)) = x₁
POL(c63(x₁)) = x₁
POL(c64(x₁)) = x₁
POL(c65(x₁)) = x₁
POL(c66(x₁)) = x₁
POL(c67(x₁)) = x₁
POL(c68(x₁)) = x₁
POL(c69(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c70(x₁)) = x₁
POL(c71(x₁)) = x₁
POL(c72(x₁)) = x₁
POL(c73(x₁)) = x₁
POL(c74(x₁)) = x₁
POL(c75(x₁)) = x₁
POL(c76(x₁)) = x₁
POL(c77(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:

U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
U92'(mark(z0)) → c21(U92'(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(mark(z0)) → c23(U62'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U42'(mark(z0)) → c30(U42'(z0))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
U51'(ok(z0)) → c42(U51'(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
X(z0, mark(z1)) → c47(X(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
U16'(ok(z0)) → c53(U16'(z0))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
S(ok(z0)) → c56(S(z0))
S(mark(z0)) → c57(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U42'(ok(z0)) → c31(U42'(z0))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(ok(z0)) → c53(U16'(z0))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U62'(mark(z0)) → c23(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
X(z0, mark(z1)) → c47(X(z0, z1))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
S(mark(z0)) → c57(S(z0))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U42'(mark(z0)) → c30(U42'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U92'(mark(z0)) → c21(U92'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(ok(z0)) → c42(U51'(z0))
S(ok(z0)) → c56(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U104', U35', U81', U31', U91', U36', ISNAT, U101', U61', U84', U92', U62', U41', PLUS, U42', U103', ISNATKIND, U82', U11', U51', U15', X, U32', U13', U16', U21', S, U23', U34', U71', U72', U12', U83', U22', U102', U14', U33', TOP

Compound Symbols:

c, c1, 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, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c70, c71, c72, c73, c74, c75, c76, c77, c3

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

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

U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))

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

U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
U92'(mark(z0)) → c21(U92'(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(mark(z0)) → c23(U62'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U42'(mark(z0)) → c30(U42'(z0))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
U51'(ok(z0)) → c42(U51'(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
X(z0, mark(z1)) → c47(X(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
U16'(ok(z0)) → c53(U16'(z0))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
S(ok(z0)) → c56(S(z0))
S(mark(z0)) → c57(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

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

POL(0) = 0
POL(ISNAT(x₁)) = 0
POL(ISNATKIND(x₁)) = 0
POL(PLUS(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(TOP(x₁)) = 0
POL(U101'(x₁, x₂, x₃)) = 0
POL(U102'(x₁, x₂, x₃)) = 0
POL(U103'(x₁, x₂, x₃)) = 0
POL(U104'(x₁, x₂, x₃)) = 0
POL(U11'(x₁, x₂, x₃)) = 0
POL(U12'(x₁, x₂, x₃)) = 0
POL(U13'(x₁, x₂, x₃)) = x₃
POL(U14'(x₁, x₂, x₃)) = x₃
POL(U15'(x₁, x₂)) = x₂
POL(U16'(x₁)) = 0
POL(U21'(x₁, x₂)) = 0
POL(U22'(x₁, x₂)) = 0
POL(U23'(x₁)) = 0
POL(U31'(x₁, x₂, x₃)) = 0
POL(U32'(x₁, x₂, x₃)) = 0
POL(U33'(x₁, x₂, x₃)) = 0
POL(U34'(x₁, x₂, x₃)) = 0
POL(U35'(x₁, x₂)) = 0
POL(U36'(x₁)) = 0
POL(U41'(x₁, x₂)) = x₁
POL(U42'(x₁)) = 0
POL(U51'(x₁)) = 0
POL(U61'(x₁, x₂)) = 0
POL(U62'(x₁)) = 0
POL(U71'(x₁, x₂)) = x₂
POL(U72'(x₁, x₂)) = 0
POL(U81'(x₁, x₂, x₃)) = x₂
POL(U82'(x₁, x₂, x₃)) = 0
POL(U83'(x₁, x₂, x₃)) = 0
POL(U84'(x₁, x₂, x₃)) = 0
POL(U91'(x₁, x₂)) = 0
POL(U92'(x₁)) = 0
POL(X(x₁, x₂)) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(c41(x₁)) = x₁
POL(c42(x₁)) = x₁
POL(c43(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(c45(x₁)) = x₁
POL(c46(x₁)) = x₁
POL(c47(x₁)) = x₁
POL(c48(x₁)) = x₁
POL(c49(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c50(x₁)) = x₁
POL(c51(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(c53(x₁)) = x₁
POL(c54(x₁)) = x₁
POL(c55(x₁)) = x₁
POL(c56(x₁)) = x₁
POL(c57(x₁)) = x₁
POL(c58(x₁)) = x₁
POL(c59(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c60(x₁)) = x₁
POL(c61(x₁)) = x₁
POL(c62(x₁)) = x₁
POL(c63(x₁)) = x₁
POL(c64(x₁)) = x₁
POL(c65(x₁)) = x₁
POL(c66(x₁)) = x₁
POL(c67(x₁)) = x₁
POL(c68(x₁)) = x₁
POL(c69(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c70(x₁)) = x₁
POL(c71(x₁)) = x₁
POL(c72(x₁)) = x₁
POL(c73(x₁)) = x₁
POL(c74(x₁)) = x₁
POL(c75(x₁)) = x₁
POL(c76(x₁)) = x₁
POL(c77(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:

U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
U92'(mark(z0)) → c21(U92'(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(mark(z0)) → c23(U62'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U42'(mark(z0)) → c30(U42'(z0))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
U51'(ok(z0)) → c42(U51'(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
X(z0, mark(z1)) → c47(X(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
U16'(ok(z0)) → c53(U16'(z0))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
S(ok(z0)) → c56(S(z0))
S(mark(z0)) → c57(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U42'(ok(z0)) → c31(U42'(z0))
U16'(ok(z0)) → c53(U16'(z0))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U62'(mark(z0)) → c23(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
X(z0, mark(z1)) → c47(X(z0, z1))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
S(mark(z0)) → c57(S(z0))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U42'(mark(z0)) → c30(U42'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U92'(mark(z0)) → c21(U92'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(ok(z0)) → c42(U51'(z0))
S(ok(z0)) → c56(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U104', U35', U81', U31', U91', U36', ISNAT, U101', U61', U84', U92', U62', U41', PLUS, U42', U103', ISNATKIND, U82', U11', U51', U15', X, U32', U13', U16', U21', S, U23', U34', U71', U72', U12', U83', U22', U102', U14', U33', TOP

Compound Symbols:

c, c1, 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, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c70, c71, c72, c73, c74, c75, c76, c77, c3

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

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

U42'(ok(z0)) → c31(U42'(z0))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))

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

U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
U92'(mark(z0)) → c21(U92'(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(mark(z0)) → c23(U62'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U42'(mark(z0)) → c30(U42'(z0))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
U51'(ok(z0)) → c42(U51'(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
X(z0, mark(z1)) → c47(X(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
U16'(ok(z0)) → c53(U16'(z0))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
S(ok(z0)) → c56(S(z0))
S(mark(z0)) → c57(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

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

POL(0) = 0
POL(ISNAT(x₁)) = x₁
POL(ISNATKIND(x₁)) = x₁
POL(PLUS(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(TOP(x₁)) = 0
POL(U101'(x₁, x₂, x₃)) = 0
POL(U102'(x₁, x₂, x₃)) = 0
POL(U103'(x₁, x₂, x₃)) = 0
POL(U104'(x₁, x₂, x₃)) = 0
POL(U11'(x₁, x₂, x₃)) = x₃
POL(U12'(x₁, x₂, x₃)) = x₂
POL(U13'(x₁, x₂, x₃)) = 0
POL(U14'(x₁, x₂, x₃)) = 0
POL(U15'(x₁, x₂)) = 0
POL(U16'(x₁)) = 0
POL(U21'(x₁, x₂)) = 0
POL(U22'(x₁, x₂)) = 0
POL(U23'(x₁)) = 0
POL(U31'(x₁, x₂, x₃)) = x₃
POL(U32'(x₁, x₂, x₃)) = 0
POL(U33'(x₁, x₂, x₃)) = 0
POL(U34'(x₁, x₂, x₃)) = 0
POL(U35'(x₁, x₂)) = 0
POL(U36'(x₁)) = 0
POL(U41'(x₁, x₂)) = 0
POL(U42'(x₁)) = x₁
POL(U51'(x₁)) = 0
POL(U61'(x₁, x₂)) = 0
POL(U62'(x₁)) = 0
POL(U71'(x₁, x₂)) = 0
POL(U72'(x₁, x₂)) = 0
POL(U81'(x₁, x₂, x₃)) = x₂
POL(U82'(x₁, x₂, x₃)) = 0
POL(U83'(x₁, x₂, x₃)) = 0
POL(U84'(x₁, x₂, x₃)) = x₃
POL(U91'(x₁, x₂)) = 0
POL(U92'(x₁)) = 0
POL(X(x₁, x₂)) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(c41(x₁)) = x₁
POL(c42(x₁)) = x₁
POL(c43(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(c45(x₁)) = x₁
POL(c46(x₁)) = x₁
POL(c47(x₁)) = x₁
POL(c48(x₁)) = x₁
POL(c49(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c50(x₁)) = x₁
POL(c51(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(c53(x₁)) = x₁
POL(c54(x₁)) = x₁
POL(c55(x₁)) = x₁
POL(c56(x₁)) = x₁
POL(c57(x₁)) = x₁
POL(c58(x₁)) = x₁
POL(c59(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c60(x₁)) = x₁
POL(c61(x₁)) = x₁
POL(c62(x₁)) = x₁
POL(c63(x₁)) = x₁
POL(c64(x₁)) = x₁
POL(c65(x₁)) = x₁
POL(c66(x₁)) = x₁
POL(c67(x₁)) = x₁
POL(c68(x₁)) = x₁
POL(c69(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c70(x₁)) = x₁
POL(c71(x₁)) = x₁
POL(c72(x₁)) = x₁
POL(c73(x₁)) = x₁
POL(c74(x₁)) = x₁
POL(c75(x₁)) = x₁
POL(c76(x₁)) = x₁
POL(c77(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

(46) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)

Tuples:

U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
U92'(mark(z0)) → c21(U92'(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(mark(z0)) → c23(U62'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U42'(mark(z0)) → c30(U42'(z0))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
U51'(ok(z0)) → c42(U51'(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
X(z0, mark(z1)) → c47(X(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
U16'(ok(z0)) → c53(U16'(z0))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
S(ok(z0)) → c56(S(z0))
S(mark(z0)) → c57(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U16'(ok(z0)) → c53(U16'(z0))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U62'(mark(z0)) → c23(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
X(z0, mark(z1)) → c47(X(z0, z1))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
S(mark(z0)) → c57(S(z0))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U42'(mark(z0)) → c30(U42'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U92'(mark(z0)) → c21(U92'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(ok(z0)) → c42(U51'(z0))
S(ok(z0)) → c56(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U42'(ok(z0)) → c31(U42'(z0))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U104', U35', U81', U31', U91', U36', ISNAT, U101', U61', U84', U92', U62', U41', PLUS, U42', U103', ISNATKIND, U82', U11', U51', U15', X, U32', U13', U16', U21', S, U23', U34', U71', U72', U12', U83', U22', U102', U14', U33', TOP

Compound Symbols:

c, c1, 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, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c70, c71, c72, c73, c74, c75, c76, c77, c3

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

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

U16'(ok(z0)) → c53(U16'(z0))

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

U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
U92'(mark(z0)) → c21(U92'(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(mark(z0)) → c23(U62'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U42'(mark(z0)) → c30(U42'(z0))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
U51'(ok(z0)) → c42(U51'(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
X(z0, mark(z1)) → c47(X(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
U16'(ok(z0)) → c53(U16'(z0))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
S(ok(z0)) → c56(S(z0))
S(mark(z0)) → c57(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

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

POL(0) = 0
POL(ISNAT(x₁)) = 0
POL(ISNATKIND(x₁)) = 0
POL(PLUS(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(TOP(x₁)) = 0
POL(U101'(x₁, x₂, x₃)) = 0
POL(U102'(x₁, x₂, x₃)) = 0
POL(U103'(x₁, x₂, x₃)) = 0
POL(U104'(x₁, x₂, x₃)) = 0
POL(U11'(x₁, x₂, x₃)) = 0
POL(U12'(x₁, x₂, x₃)) = 0
POL(U13'(x₁, x₂, x₃)) = 0
POL(U14'(x₁, x₂, x₃)) = 0
POL(U15'(x₁, x₂)) = 0
POL(U16'(x₁)) = x₁
POL(U21'(x₁, x₂)) = 0
POL(U22'(x₁, x₂)) = 0
POL(U23'(x₁)) = 0
POL(U31'(x₁, x₂, x₃)) = 0
POL(U32'(x₁, x₂, x₃)) = 0
POL(U33'(x₁, x₂, x₃)) = 0
POL(U34'(x₁, x₂, x₃)) = 0
POL(U35'(x₁, x₂)) = 0
POL(U36'(x₁)) = 0
POL(U41'(x₁, x₂)) = 0
POL(U42'(x₁)) = 0
POL(U51'(x₁)) = 0
POL(U61'(x₁, x₂)) = 0
POL(U62'(x₁)) = 0
POL(U71'(x₁, x₂)) = 0
POL(U72'(x₁, x₂)) = 0
POL(U81'(x₁, x₂, x₃)) = 0
POL(U82'(x₁, x₂, x₃)) = 0
POL(U83'(x₁, x₂, x₃)) = 0
POL(U84'(x₁, x₂, x₃)) = 0
POL(U91'(x₁, x₂)) = 0
POL(U92'(x₁)) = 0
POL(X(x₁, x₂)) = 0
POL(c(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c17(x₁)) = x₁
POL(c18(x₁)) = x₁
POL(c19(x₁)) = x₁
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(c41(x₁)) = x₁
POL(c42(x₁)) = x₁
POL(c43(x₁)) = x₁
POL(c44(x₁)) = x₁
POL(c45(x₁)) = x₁
POL(c46(x₁)) = x₁
POL(c47(x₁)) = x₁
POL(c48(x₁)) = x₁
POL(c49(x₁)) = x₁
POL(c5(x₁)) = x₁
POL(c50(x₁)) = x₁
POL(c51(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(c53(x₁)) = x₁
POL(c54(x₁)) = x₁
POL(c55(x₁)) = x₁
POL(c56(x₁)) = x₁
POL(c57(x₁)) = x₁
POL(c58(x₁)) = x₁
POL(c59(x₁)) = x₁
POL(c6(x₁)) = x₁
POL(c60(x₁)) = x₁
POL(c61(x₁)) = x₁
POL(c62(x₁)) = x₁
POL(c63(x₁)) = x₁
POL(c64(x₁)) = x₁
POL(c65(x₁)) = x₁
POL(c66(x₁)) = x₁
POL(c67(x₁)) = x₁
POL(c68(x₁)) = x₁
POL(c69(x₁)) = x₁
POL(c7(x₁)) = x₁
POL(c70(x₁)) = x₁
POL(c71(x₁)) = x₁
POL(c72(x₁)) = x₁
POL(c73(x₁)) = x₁
POL(c74(x₁)) = x₁
POL(c75(x₁)) = x₁
POL(c76(x₁)) = x₁
POL(c77(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

(48) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(0) → ok(0)

Tuples:

U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
U92'(mark(z0)) → c21(U92'(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(mark(z0)) → c23(U62'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U42'(mark(z0)) → c30(U42'(z0))
U42'(ok(z0)) → c31(U42'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
U51'(ok(z0)) → c42(U51'(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
X(z0, mark(z1)) → c47(X(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
U16'(ok(z0)) → c53(U16'(z0))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
S(ok(z0)) → c56(S(z0))
S(mark(z0)) → c57(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:none
K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
U31'(ok(z0), ok(z1), ok(z2)) → c8(U31'(z0, z1, z2))
U31'(mark(z0), z1, z2) → c9(U31'(z0, z1, z2))
U91'(ok(z0), ok(z1)) → c11(U91'(z0, z1))
U36'(mark(z0)) → c12(U36'(z0))
U36'(ok(z0)) → c13(U36'(z0))
U84'(ok(z0), ok(z1), ok(z2)) → c20(U84'(z0, z1, z2))
PLUS(ok(z0), ok(z1)) → c27(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c28(PLUS(z0, z1))
U32'(ok(z0), ok(z1), ok(z2)) → c48(U32'(z0, z1, z2))
U21'(ok(z0), ok(z1)) → c54(U21'(z0, z1))
U21'(mark(z0), z1) → c55(U21'(z0, z1))
U34'(ok(z0), ok(z1), ok(z2)) → c60(U34'(z0, z1, z2))
U34'(mark(z0), z1, z2) → c61(U34'(z0, z1, z2))
U33'(mark(z0), z1, z2) → c76(U33'(z0, z1, z2))
U33'(ok(z0), ok(z1), ok(z2)) → c77(U33'(z0, z1, z2))
U35'(ok(z0), ok(z1)) → c4(U35'(z0, z1))
U71'(ok(z0), ok(z1)) → c62(U71'(z0, z1))
U104'(mark(z0), z1, z2) → c1(U104'(z0, z1, z2))
U81'(mark(z0), z1, z2) → c6(U81'(z0, z1, z2))
U84'(mark(z0), z1, z2) → c19(U84'(z0, z1, z2))
U62'(mark(z0)) → c23(U62'(z0))
U41'(mark(z0), z1) → c25(U41'(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
U82'(mark(z0), z1, z2) → c38(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c39(U11'(z0, z1, z2))
U51'(mark(z0)) → c41(U51'(z0))
X(z0, mark(z1)) → c47(X(z0, z1))
U13'(mark(z0), z1, z2) → c50(U13'(z0, z1, z2))
U16'(mark(z0)) → c52(U16'(z0))
S(mark(z0)) → c57(S(z0))
U71'(mark(z0), z1) → c63(U71'(z0, z1))
U12'(mark(z0), z1, z2) → c67(U12'(z0, z1, z2))
U83'(mark(z0), z1, z2) → c69(U83'(z0, z1, z2))
U22'(mark(z0), z1) → c70(U22'(z0, z1))
U102'(mark(z0), z1, z2) → c72(U102'(z0, z1, z2))
U91'(mark(z0), z1) → c10(U91'(z0, z1))
U101'(mark(z0), z1, z2) → c16(U101'(z0, z1, z2))
U42'(mark(z0)) → c30(U42'(z0))
U23'(mark(z0)) → c59(U23'(z0))
U14'(mark(z0), z1, z2) → c75(U14'(z0, z1, z2))
U32'(mark(z0), z1, z2) → c49(U32'(z0, z1, z2))
U15'(mark(z0), z1) → c44(U15'(z0, z1))
X(mark(z0), z1) → c46(X(z0, z1))
U104'(ok(z0), ok(z1), ok(z2)) → c(U104'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c7(U81'(z0, z1, z2))
U101'(ok(z0), ok(z1), ok(z2)) → c15(U101'(z0, z1, z2))
U61'(ok(z0), ok(z1)) → c17(U61'(z0, z1))
U41'(ok(z0), ok(z1)) → c26(U41'(z0, z1))
U103'(ok(z0), ok(z1), ok(z2)) → c33(U103'(z0, z1, z2))
ISNATKIND(ok(z0)) → c36(ISNATKIND(z0))
U15'(ok(z0), ok(z1)) → c43(U15'(z0, z1))
X(ok(z0), ok(z1)) → c45(X(z0, z1))
U83'(ok(z0), ok(z1), ok(z2)) → c68(U83'(z0, z1, z2))
U14'(ok(z0), ok(z1), ok(z2)) → c74(U14'(z0, z1, z2))
U35'(mark(z0), z1) → c5(U35'(z0, z1))
U72'(mark(z0), z1) → c64(U72'(z0, z1))
U92'(mark(z0)) → c21(U92'(z0))
U103'(mark(z0), z1, z2) → c32(U103'(z0, z1, z2))
U61'(mark(z0), z1) → c18(U61'(z0, z1))
U72'(ok(z0), ok(z1)) → c65(U72'(z0, z1))
ISNAT(ok(z0)) → c14(ISNAT(z0))
U92'(ok(z0)) → c22(U92'(z0))
U62'(ok(z0)) → c24(U62'(z0))
U82'(ok(z0), ok(z1), ok(z2)) → c37(U82'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c40(U11'(z0, z1, z2))
U51'(ok(z0)) → c42(U51'(z0))
S(ok(z0)) → c56(S(z0))
U23'(ok(z0)) → c58(U23'(z0))
U22'(ok(z0), ok(z1)) → c71(U22'(z0, z1))
U102'(ok(z0), ok(z1), ok(z2)) → c73(U102'(z0, z1, z2))
U13'(ok(z0), ok(z1), ok(z2)) → c51(U13'(z0, z1, z2))
U42'(ok(z0)) → c31(U42'(z0))
U12'(ok(z0), ok(z1), ok(z2)) → c66(U12'(z0, z1, z2))
U16'(ok(z0)) → c53(U16'(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U104', U35', U81', U31', U91', U36', ISNAT, U101', U61', U84', U92', U62', U41', PLUS, U42', U103', ISNATKIND, U82', U11', U51', U15', X, U32', U13', U16', U21', S, U23', U34', U71', U72', U12', U83', U22', U102', U14', U33', TOP

Compound Symbols:

c, c1, 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, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c70, c71, c72, c73, c74, c75, c76, c77, c3

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

The set S is empty

(0) Obligation:

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

(2) Obligation:

(3) CpxTrsMatchBoundsTAProof (EQUIVALENT transformation)

(4) BOUNDS(1, n^1)

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

(6) Obligation:

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

(8) Obligation:

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

(10) Obligation:

(11) CdtUsableRulesProof (EQUIVALENT transformation)

(12) Obligation:

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

(14) Obligation:

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

(16) Obligation:

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

(18) Obligation:

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

(20) Obligation:

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

(22) Obligation:

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

(24) Obligation:

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

(26) Obligation:

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

(28) Obligation:

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

(30) Obligation:

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

(32) Obligation:

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

(34) Obligation:

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

(36) Obligation:

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

(38) Obligation:

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

(40) Obligation:

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

(42) Obligation:

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

(44) Obligation:

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

(46) Obligation:

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

(48) Obligation:

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

(50) BOUNDS(1, 1)