Runtime Complexity TRS:
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

Renamed function symbols to avoid clashes with predefined symbol.

Runtime Complexity TRS:
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

Infered types.

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))

Types:
active' :: tt':mark':0':ok' → tt':mark':0':ok'
U101' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
tt' :: tt':mark':0':ok'
mark' :: tt':mark':0':ok' → tt':mark':0':ok'
U102' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
isNatKind' :: tt':mark':0':ok' → tt':mark':0':ok'
U103' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
isNat' :: tt':mark':0':ok' → tt':mark':0':ok'
U104' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
plus' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
x' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
U11' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
U12' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
U13' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
U14' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
U15' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
U16' :: tt':mark':0':ok' → tt':mark':0':ok'
U21' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
U22' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
U23' :: tt':mark':0':ok' → tt':mark':0':ok'
U31' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
U32' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
U33' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
U34' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
U35' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
U36' :: tt':mark':0':ok' → tt':mark':0':ok'
U41' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
U42' :: tt':mark':0':ok' → tt':mark':0':ok'
U51' :: tt':mark':0':ok' → tt':mark':0':ok'
U61' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
U62' :: tt':mark':0':ok' → tt':mark':0':ok'
U71' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
U72' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
U81' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
U82' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
U83' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
U84' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
s' :: tt':mark':0':ok' → tt':mark':0':ok'
U91' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
U92' :: tt':mark':0':ok' → tt':mark':0':ok'
0' :: tt':mark':0':ok'
proper' :: tt':mark':0':ok' → tt':mark':0':ok'
ok' :: tt':mark':0':ok' → tt':mark':0':ok'
top' :: tt':mark':0':ok' → top'
_hole_tt':mark':0':ok'1 :: tt':mark':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':0':ok'3 :: Nat → tt':mark':0':ok'

Heuristically decided to analyse the following defined symbols:
active', U102', isNatKind', U103', isNat', U104', plus', x', U12', U13', U14', U15', U16', U22', U23', U32', U33', U34', U35', U36', U42', U62', U72', U82', U83', U84', s', U92', U11', U21', U31', U41', U51', U61', U71', U81', U91', U101', proper', top'

They will be analysed ascendingly in the following order:
U102' < active'
isNatKind' < active'
U103' < active'
isNat' < active'
U104' < active'
plus' < active'
x' < active'
U12' < active'
U13' < active'
U14' < active'
U15' < active'
U16' < active'
U22' < active'
U23' < active'
U32' < active'
U33' < active'
U34' < active'
U35' < active'
U36' < active'
U42' < active'
U62' < active'
U72' < active'
U82' < active'
U83' < active'
U84' < active'
s' < active'
U92' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
U91' < active'
U101' < active'
active' < top'
U102' < proper'
isNatKind' < proper'
U103' < proper'
isNat' < proper'
U104' < proper'
plus' < proper'
x' < proper'
U12' < proper'
U13' < proper'
U14' < proper'
U15' < proper'
U16' < proper'
U22' < proper'
U23' < proper'
U32' < proper'
U33' < proper'
U34' < proper'
U35' < proper'
U36' < proper'
U42' < proper'
U62' < proper'
U72' < proper'
U82' < proper'
U83' < proper'
U84' < proper'
s' < proper'
U92' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
U71' < proper'
U81' < proper'
U91' < proper'
U101' < proper'
proper' < top'

Generator Equations:
_gen_tt':mark':0':ok'3(0) ⇔ tt'
_gen_tt':mark':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':ok'3(x))

The following defined symbols remain to be analysed:
U102', active', isNatKind', U103', isNat', U104', plus', x', U12', U13', U14', U15', U16', U22', U23', U32', U33', U34', U35', U36', U42', U62', U72', U82', U83', U84', s', U92', U11', U21', U31', U41', U51', U61', U71', U81', U91', U101', proper', top'

Proved the following rewrite lemma:
U102'(_gen_tt':mark':0':ok'3(+(1, _n5)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(_n5)

Induction Base:
U102'(_gen_tt':mark':0':ok'3(+(1, 0)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c))

Induction Step:
U102'(_gen_tt':mark':0':ok'3(+(1, +(_$n6, 1))), _gen_tt':mark':0':ok'3(_b1018), _gen_tt':mark':0':ok'3(_c1019)) →_R^Ω(1)
mark'(U102'(_gen_tt':mark':0':ok'3(+(1, _$n6)), _gen_tt':mark':0':ok'3(_b1018), _gen_tt':mark':0':ok'3(_c1019))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Lemmas:
U102'(_gen_tt':mark':0':ok'3(+(1, _n5)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(_n5)

Generator Equations:
_gen_tt':mark':0':ok'3(0) ⇔ tt'
_gen_tt':mark':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':ok'3(x))

The following defined symbols remain to be analysed:
isNatKind', active', U103', isNat', U104', plus', x', U12', U13', U14', U15', U16', U22', U23', U32', U33', U34', U35', U36', U42', U62', U72', U82', U83', U84', s', U92', U11', U21', U31', U41', U51', U61', U71', U81', U91', U101', proper', top'

They will be analysed ascendingly in the following order:
isNatKind' < active'
U103' < active'
isNat' < active'
U104' < active'
plus' < active'
x' < active'
U12' < active'
U13' < active'
U14' < active'
U15' < active'
U16' < active'
U22' < active'
U23' < active'
U32' < active'
U33' < active'
U34' < active'
U35' < active'
U36' < active'
U42' < active'
U62' < active'
U72' < active'
U82' < active'
U83' < active'
U84' < active'
s' < active'
U92' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
U91' < active'
U101' < active'
active' < top'
isNatKind' < proper'
U103' < proper'
isNat' < proper'
U104' < proper'
plus' < proper'
x' < proper'
U12' < proper'
U13' < proper'
U14' < proper'
U15' < proper'
U16' < proper'
U22' < proper'
U23' < proper'
U32' < proper'
U33' < proper'
U34' < proper'
U35' < proper'
U36' < proper'
U42' < proper'
U62' < proper'
U72' < proper'
U82' < proper'
U83' < proper'
U84' < proper'
s' < proper'
U92' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
U71' < proper'
U81' < proper'
U91' < proper'
U101' < proper'
proper' < top'

Could not prove a rewrite lemma for the defined symbol isNatKind'.

Lemmas:
U102'(_gen_tt':mark':0':ok'3(+(1, _n5)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(_n5)

Generator Equations:
_gen_tt':mark':0':ok'3(0) ⇔ tt'
_gen_tt':mark':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':ok'3(x))

The following defined symbols remain to be analysed:
U103', active', isNat', U104', plus', x', U12', U13', U14', U15', U16', U22', U23', U32', U33', U34', U35', U36', U42', U62', U72', U82', U83', U84', s', U92', U11', U21', U31', U41', U51', U61', U71', U81', U91', U101', proper', top'

They will be analysed ascendingly in the following order:
U103' < active'
isNat' < active'
U104' < active'
plus' < active'
x' < active'
U12' < active'
U13' < active'
U14' < active'
U15' < active'
U16' < active'
U22' < active'
U23' < active'
U32' < active'
U33' < active'
U34' < active'
U35' < active'
U36' < active'
U42' < active'
U62' < active'
U72' < active'
U82' < active'
U83' < active'
U84' < active'
s' < active'
U92' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
U91' < active'
U101' < active'
active' < top'
U103' < proper'
isNat' < proper'
U104' < proper'
plus' < proper'
x' < proper'
U12' < proper'
U13' < proper'
U14' < proper'
U15' < proper'
U16' < proper'
U22' < proper'
U23' < proper'
U32' < proper'
U33' < proper'
U34' < proper'
U35' < proper'
U36' < proper'
U42' < proper'
U62' < proper'
U72' < proper'
U82' < proper'
U83' < proper'
U84' < proper'
s' < proper'
U92' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
U71' < proper'
U81' < proper'
U91' < proper'
U101' < proper'
proper' < top'

Proved the following rewrite lemma:
U103'(_gen_tt':mark':0':ok'3(+(1, _n14924)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(_n14924)

Induction Base:
U103'(_gen_tt':mark':0':ok'3(+(1, 0)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c))

Induction Step:
U103'(_gen_tt':mark':0':ok'3(+(1, +(_$n14925, 1))), _gen_tt':mark':0':ok'3(_b16585), _gen_tt':mark':0':ok'3(_c16586)) →_R^Ω(1)
mark'(U103'(_gen_tt':mark':0':ok'3(+(1, _$n14925)), _gen_tt':mark':0':ok'3(_b16585), _gen_tt':mark':0':ok'3(_c16586))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_tt':mark':0':ok'3(0) ⇔ tt'
_gen_tt':mark':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':ok'3(x))

The following defined symbols remain to be analysed:
isNat', active', U104', plus', x', U12', U13', U14', U15', U16', U22', U23', U32', U33', U34', U35', U36', U42', U62', U72', U82', U83', U84', s', U92', U11', U21', U31', U41', U51', U61', U71', U81', U91', U101', proper', top'

They will be analysed ascendingly in the following order:
isNat' < active'
U104' < active'
plus' < active'
x' < active'
U12' < active'
U13' < active'
U14' < active'
U15' < active'
U16' < active'
U22' < active'
U23' < active'
U32' < active'
U33' < active'
U34' < active'
U35' < active'
U36' < active'
U42' < active'
U62' < active'
U72' < active'
U82' < active'
U83' < active'
U84' < active'
s' < active'
U92' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
U91' < active'
U101' < active'
active' < top'
isNat' < proper'
U104' < proper'
plus' < proper'
x' < proper'
U12' < proper'
U13' < proper'
U14' < proper'
U15' < proper'
U16' < proper'
U22' < proper'
U23' < proper'
U32' < proper'
U33' < proper'
U34' < proper'
U35' < proper'
U36' < proper'
U42' < proper'
U62' < proper'
U72' < proper'
U82' < proper'
U83' < proper'
U84' < proper'
s' < proper'
U92' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
U71' < proper'
U81' < proper'
U91' < proper'
U101' < proper'
proper' < top'

Could not prove a rewrite lemma for the defined symbol isNat'.

Generator Equations:
_gen_tt':mark':0':ok'3(0) ⇔ tt'
_gen_tt':mark':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':ok'3(x))

The following defined symbols remain to be analysed:
U104', active', plus', x', U12', U13', U14', U15', U16', U22', U23', U32', U33', U34', U35', U36', U42', U62', U72', U82', U83', U84', s', U92', U11', U21', U31', U41', U51', U61', U71', U81', U91', U101', proper', top'

They will be analysed ascendingly in the following order:
U104' < active'
plus' < active'
x' < active'
U12' < active'
U13' < active'
U14' < active'
U15' < active'
U16' < active'
U22' < active'
U23' < active'
U32' < active'
U33' < active'
U34' < active'
U35' < active'
U36' < active'
U42' < active'
U62' < active'
U72' < active'
U82' < active'
U83' < active'
U84' < active'
s' < active'
U92' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
U91' < active'
U101' < active'
active' < top'
U104' < proper'
plus' < proper'
x' < proper'
U12' < proper'
U13' < proper'
U14' < proper'
U15' < proper'
U16' < proper'
U22' < proper'
U23' < proper'
U32' < proper'
U33' < proper'
U34' < proper'
U35' < proper'
U36' < proper'
U42' < proper'
U62' < proper'
U72' < proper'
U82' < proper'
U83' < proper'
U84' < proper'
s' < proper'
U92' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
U71' < proper'
U81' < proper'
U91' < proper'
U101' < proper'
proper' < top'

Proved the following rewrite lemma:
U104'(_gen_tt':mark':0':ok'3(+(1, _n30589)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(_n30589)

Induction Base:
U104'(_gen_tt':mark':0':ok'3(+(1, 0)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c))

Induction Step:
U104'(_gen_tt':mark':0':ok'3(+(1, +(_$n30590, 1))), _gen_tt':mark':0':ok'3(_b32898), _gen_tt':mark':0':ok'3(_c32899)) →_R^Ω(1)
mark'(U104'(_gen_tt':mark':0':ok'3(+(1, _$n30590)), _gen_tt':mark':0':ok'3(_b32898), _gen_tt':mark':0':ok'3(_c32899))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_tt':mark':0':ok'3(0) ⇔ tt'
_gen_tt':mark':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':ok'3(x))

The following defined symbols remain to be analysed:
plus', active', x', U12', U13', U14', U15', U16', U22', U23', U32', U33', U34', U35', U36', U42', U62', U72', U82', U83', U84', s', U92', U11', U21', U31', U41', U51', U61', U71', U81', U91', U101', proper', top'

They will be analysed ascendingly in the following order:
plus' < active'
x' < active'
U12' < active'
U13' < active'
U14' < active'
U15' < active'
U16' < active'
U22' < active'
U23' < active'
U32' < active'
U33' < active'
U34' < active'
U35' < active'
U36' < active'
U42' < active'
U62' < active'
U72' < active'
U82' < active'
U83' < active'
U84' < active'
s' < active'
U92' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
U91' < active'
U101' < active'
active' < top'
plus' < proper'
x' < proper'
U12' < proper'
U13' < proper'
U14' < proper'
U15' < proper'
U16' < proper'
U22' < proper'
U23' < proper'
U32' < proper'
U33' < proper'
U34' < proper'
U35' < proper'
U36' < proper'
U42' < proper'
U62' < proper'
U72' < proper'
U82' < proper'
U83' < proper'
U84' < proper'
s' < proper'
U92' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
U71' < proper'
U81' < proper'
U91' < proper'
U101' < proper'
proper' < top'

Proved the following rewrite lemma:
plus'(_gen_tt':mark':0':ok'3(+(1, _n46952)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(_n46952)

Induction Base:
plus'(_gen_tt':mark':0':ok'3(+(1, 0)), _gen_tt':mark':0':ok'3(b))

Induction Step:
plus'(_gen_tt':mark':0':ok'3(+(1, +(_$n46953, 1))), _gen_tt':mark':0':ok'3(_b49069)) →_R^Ω(1)
mark'(plus'(_gen_tt':mark':0':ok'3(+(1, _$n46953)), _gen_tt':mark':0':ok'3(_b49069))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_tt':mark':0':ok'3(0) ⇔ tt'
_gen_tt':mark':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':ok'3(x))

The following defined symbols remain to be analysed:
x', active', U12', U13', U14', U15', U16', U22', U23', U32', U33', U34', U35', U36', U42', U62', U72', U82', U83', U84', s', U92', U11', U21', U31', U41', U51', U61', U71', U81', U91', U101', proper', top'

They will be analysed ascendingly in the following order:
x' < active'
U12' < active'
U13' < active'
U14' < active'
U15' < active'
U16' < active'
U22' < active'
U23' < active'
U32' < active'
U33' < active'
U34' < active'
U35' < active'
U36' < active'
U42' < active'
U62' < active'
U72' < active'
U82' < active'
U83' < active'
U84' < active'
s' < active'
U92' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
U91' < active'
U101' < active'
active' < top'
x' < proper'
U12' < proper'
U13' < proper'
U14' < proper'
U15' < proper'
U16' < proper'
U22' < proper'
U23' < proper'
U32' < proper'
U33' < proper'
U34' < proper'
U35' < proper'
U36' < proper'
U42' < proper'
U62' < proper'
U72' < proper'
U82' < proper'
U83' < proper'
U84' < proper'
s' < proper'
U92' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
U71' < proper'
U81' < proper'
U91' < proper'
U101' < proper'
proper' < top'

Proved the following rewrite lemma:
x'(_gen_tt':mark':0':ok'3(+(1, _n59663)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(_n59663)

Induction Base:
x'(_gen_tt':mark':0':ok'3(+(1, 0)), _gen_tt':mark':0':ok'3(b))

Induction Step:
x'(_gen_tt':mark':0':ok'3(+(1, +(_$n59664, 1))), _gen_tt':mark':0':ok'3(_b62104)) →_R^Ω(1)
mark'(x'(_gen_tt':mark':0':ok'3(+(1, _$n59664)), _gen_tt':mark':0':ok'3(_b62104))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Lemmas:
U102'(_gen_tt':mark':0':ok'3(+(1, _n5)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(_n5)
U103'(_gen_tt':mark':0':ok'3(+(1, _n14924)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(_n14924)
U104'(_gen_tt':mark':0':ok'3(+(1, _n30589)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(_n30589)
plus'(_gen_tt':mark':0':ok'3(+(1, _n46952)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(_n46952)
x'(_gen_tt':mark':0':ok'3(+(1, _n59663)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(_n59663)

Generator Equations:
_gen_tt':mark':0':ok'3(0) ⇔ tt'
_gen_tt':mark':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':ok'3(x))

The following defined symbols remain to be analysed:
U12', active', U13', U14', U15', U16', U22', U23', U32', U33', U34', U35', U36', U42', U62', U72', U82', U83', U84', s', U92', U11', U21', U31', U41', U51', U61', U71', U81', U91', U101', proper', top'

They will be analysed ascendingly in the following order:
U12' < active'
U13' < active'
U14' < active'
U15' < active'
U16' < active'
U22' < active'
U23' < active'
U32' < active'
U33' < active'
U34' < active'
U35' < active'
U36' < active'
U42' < active'
U62' < active'
U72' < active'
U82' < active'
U83' < active'
U84' < active'
s' < active'
U92' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
U91' < active'
U101' < active'
active' < top'
U12' < proper'
U13' < proper'
U14' < proper'
U15' < proper'
U16' < proper'
U22' < proper'
U23' < proper'
U32' < proper'
U33' < proper'
U34' < proper'
U35' < proper'
U36' < proper'
U42' < proper'
U62' < proper'
U72' < proper'
U82' < proper'
U83' < proper'
U84' < proper'
s' < proper'
U92' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
U71' < proper'
U81' < proper'
U91' < proper'
U101' < proper'
proper' < top'

Proved the following rewrite lemma:
U12'(_gen_tt':mark':0':ok'3(+(1, _n72742)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(_n72742)

Induction Base:
U12'(_gen_tt':mark':0':ok'3(+(1, 0)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c))

Induction Step:
U12'(_gen_tt':mark':0':ok'3(+(1, +(_$n72743, 1))), _gen_tt':mark':0':ok'3(_b76671), _gen_tt':mark':0':ok'3(_c76672)) →_R^Ω(1)
mark'(U12'(_gen_tt':mark':0':ok'3(+(1, _$n72743)), _gen_tt':mark':0':ok'3(_b76671), _gen_tt':mark':0':ok'3(_c76672))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_tt':mark':0':ok'3(0) ⇔ tt'
_gen_tt':mark':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':ok'3(x))

The following defined symbols remain to be analysed:
U13', active', U14', U15', U16', U22', U23', U32', U33', U34', U35', U36', U42', U62', U72', U82', U83', U84', s', U92', U11', U21', U31', U41', U51', U61', U71', U81', U91', U101', proper', top'

They will be analysed ascendingly in the following order:
U13' < active'
U14' < active'
U15' < active'
U16' < active'
U22' < active'
U23' < active'
U32' < active'
U33' < active'
U34' < active'
U35' < active'
U36' < active'
U42' < active'
U62' < active'
U72' < active'
U82' < active'
U83' < active'
U84' < active'
s' < active'
U92' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
U91' < active'
U101' < active'
active' < top'
U13' < proper'
U14' < proper'
U15' < proper'
U16' < proper'
U22' < proper'
U23' < proper'
U32' < proper'
U33' < proper'
U34' < proper'
U35' < proper'
U36' < proper'
U42' < proper'
U62' < proper'
U72' < proper'
U82' < proper'
U83' < proper'
U84' < proper'
s' < proper'
U92' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
U71' < proper'
U81' < proper'
U91' < proper'
U101' < proper'
proper' < top'

Proved the following rewrite lemma:
U13'(_gen_tt':mark':0':ok'3(+(1, _n90927)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(_n90927)

Induction Base:
U13'(_gen_tt':mark':0':ok'3(+(1, 0)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c))

Induction Step:
U13'(_gen_tt':mark':0':ok'3(+(1, +(_$n90928, 1))), _gen_tt':mark':0':ok'3(_b95504), _gen_tt':mark':0':ok'3(_c95505)) →_R^Ω(1)
mark'(U13'(_gen_tt':mark':0':ok'3(+(1, _$n90928)), _gen_tt':mark':0':ok'3(_b95504), _gen_tt':mark':0':ok'3(_c95505))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).