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

0 CpxTRS
↳1 NestedDefinedSymbolProof (BOTH BOUNDS(ID, ID), 14 ms)
↳2 CpxTRS
↳3 RcToIrcProof (BOTH BOUNDS(ID, ID), 134 ms)
↳4 CpxTRS
↳5 CpxTrsMatchBoundsTAProof (⇔, 2622 ms)
↳6 BOUNDS(1, n^1)

(0) Obligation:

The Runtime Complexity (full) 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: FULL

(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 (full) 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: FULL

(3) RcToIrcProof (BOTH BOUNDS(ID, ID) transformation)

Converted rc-obligation to irc-obligation.

As the TRS is a non-duplicating overlay system, we have rc = irc.

(4) 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

(5) 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

(6) BOUNDS(1, n^1)