Runtime Complexity TRS:
The TRS R consists of the following rules:

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, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
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)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X)) → U41(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2, X3)) → U62(active(X1), X2, X3)
active(U63(X1, X2, X3)) → U63(active(X1), X2, X3)
active(U64(X1, X2, X3)) → U64(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
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) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X)) → mark(U41(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2, X3) → mark(U62(X1, X2, X3))
U63(mark(X1), X2, X3) → mark(U63(X1, X2, X3))
U64(mark(X1), X2, X3) → mark(U64(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
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(isNat(X)) → isNat(proper(X))
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)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X)) → U41(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2, X3)) → U62(proper(X1), proper(X2), proper(X3))
proper(U63(X1, X2, X3)) → U63(proper(X1), proper(X2), proper(X3))
proper(U64(X1, X2, X3)) → U64(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0) → ok(0)
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
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))
isNat(ok(X)) → ok(isNat(X))
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(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X)) → ok(U41(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2), ok(X3)) → ok(U62(X1, X2, X3))
U63(ok(X1), ok(X2), ok(X3)) → ok(U63(X1, X2, X3))
U64(ok(X1), ok(X2), ok(X3)) → ok(U64(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
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'(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', V2)) → mark'(U32'(isNatKind'(V2)))
active'(U32'(tt')) → mark'(tt')
active'(U41'(tt')) → mark'(tt')
active'(U51'(tt', N)) → mark'(U52'(isNatKind'(N), N))
active'(U52'(tt', N)) → mark'(N)
active'(U61'(tt', M, N)) → mark'(U62'(isNatKind'(M), M, N))
active'(U62'(tt', M, N)) → mark'(U63'(isNat'(N), M, N))
active'(U63'(tt', M, N)) → mark'(U64'(isNatKind'(N), M, N))
active'(U64'(tt', M, N)) → mark'(s'(plus'(N, M)))
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'(isNatKind'(0')) → mark'(tt')
active'(isNatKind'(plus'(V1, V2))) → mark'(U31'(isNatKind'(V1), V2))
active'(isNatKind'(s'(V1))) → mark'(U41'(isNatKind'(V1)))
active'(plus'(N, 0')) → mark'(U51'(isNat'(N), N))
active'(plus'(N, s'(M))) → mark'(U61'(isNat'(M), M, N))
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)) → U31'(active'(X1), X2)
active'(U32'(X)) → U32'(active'(X))
active'(U41'(X)) → U41'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X1, X2)) → U52'(active'(X1), X2)
active'(U61'(X1, X2, X3)) → U61'(active'(X1), X2, X3)
active'(U62'(X1, X2, X3)) → U62'(active'(X1), X2, X3)
active'(U63'(X1, X2, X3)) → U63'(active'(X1), X2, X3)
active'(U64'(X1, X2, X3)) → U64'(active'(X1), X2, X3)
active'(s'(X)) → s'(active'(X))
active'(plus'(X1, X2)) → plus'(active'(X1), X2)
active'(plus'(X1, X2)) → plus'(X1, active'(X2))
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) → mark'(U31'(X1, X2))
U32'(mark'(X)) → mark'(U32'(X))
U41'(mark'(X)) → mark'(U41'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X1), X2) → mark'(U52'(X1, X2))
U61'(mark'(X1), X2, X3) → mark'(U61'(X1, X2, X3))
U62'(mark'(X1), X2, X3) → mark'(U62'(X1, X2, X3))
U63'(mark'(X1), X2, X3) → mark'(U63'(X1, X2, X3))
U64'(mark'(X1), X2, X3) → mark'(U64'(X1, X2, X3))
s'(mark'(X)) → mark'(s'(X))
plus'(mark'(X1), X2) → mark'(plus'(X1, X2))
plus'(X1, mark'(X2)) → mark'(plus'(X1, X2))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(tt') → ok'(tt')
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(isNatKind'(X)) → isNatKind'(proper'(X))
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'(isNat'(X)) → isNat'(proper'(X))
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)) → U31'(proper'(X1), proper'(X2))
proper'(U32'(X)) → U32'(proper'(X))
proper'(U41'(X)) → U41'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X1, X2)) → U52'(proper'(X1), proper'(X2))
proper'(U61'(X1, X2, X3)) → U61'(proper'(X1), proper'(X2), proper'(X3))
proper'(U62'(X1, X2, X3)) → U62'(proper'(X1), proper'(X2), proper'(X3))
proper'(U63'(X1, X2, X3)) → U63'(proper'(X1), proper'(X2), proper'(X3))
proper'(U64'(X1, X2, X3)) → U64'(proper'(X1), proper'(X2), proper'(X3))
proper'(s'(X)) → s'(proper'(X))
proper'(plus'(X1, X2)) → plus'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
isNatKind'(ok'(X)) → ok'(isNatKind'(X))
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))
isNat'(ok'(X)) → ok'(isNat'(X))
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'(U31'(X1, X2))
U32'(ok'(X)) → ok'(U32'(X))
U41'(ok'(X)) → ok'(U41'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X1), ok'(X2)) → ok'(U52'(X1, X2))
U61'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U61'(X1, X2, X3))
U62'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U62'(X1, X2, X3))
U63'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U63'(X1, X2, X3))
U64'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U64'(X1, X2, X3))
s'(ok'(X)) → ok'(s'(X))
plus'(ok'(X1), ok'(X2)) → ok'(plus'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Rewrite Strategy: INNERMOST

Infered types.

Rules:
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', V2)) → mark'(U32'(isNatKind'(V2)))
active'(U32'(tt')) → mark'(tt')
active'(U41'(tt')) → mark'(tt')
active'(U51'(tt', N)) → mark'(U52'(isNatKind'(N), N))
active'(U52'(tt', N)) → mark'(N)
active'(U61'(tt', M, N)) → mark'(U62'(isNatKind'(M), M, N))
active'(U62'(tt', M, N)) → mark'(U63'(isNat'(N), M, N))
active'(U63'(tt', M, N)) → mark'(U64'(isNatKind'(N), M, N))
active'(U64'(tt', M, N)) → mark'(s'(plus'(N, M)))
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'(isNatKind'(0')) → mark'(tt')
active'(isNatKind'(plus'(V1, V2))) → mark'(U31'(isNatKind'(V1), V2))
active'(isNatKind'(s'(V1))) → mark'(U41'(isNatKind'(V1)))
active'(plus'(N, 0')) → mark'(U51'(isNat'(N), N))
active'(plus'(N, s'(M))) → mark'(U61'(isNat'(M), M, N))
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)) → U31'(active'(X1), X2)
active'(U32'(X)) → U32'(active'(X))
active'(U41'(X)) → U41'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X1, X2)) → U52'(active'(X1), X2)
active'(U61'(X1, X2, X3)) → U61'(active'(X1), X2, X3)
active'(U62'(X1, X2, X3)) → U62'(active'(X1), X2, X3)
active'(U63'(X1, X2, X3)) → U63'(active'(X1), X2, X3)
active'(U64'(X1, X2, X3)) → U64'(active'(X1), X2, X3)
active'(s'(X)) → s'(active'(X))
active'(plus'(X1, X2)) → plus'(active'(X1), X2)
active'(plus'(X1, X2)) → plus'(X1, active'(X2))
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) → mark'(U31'(X1, X2))
U32'(mark'(X)) → mark'(U32'(X))
U41'(mark'(X)) → mark'(U41'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X1), X2) → mark'(U52'(X1, X2))
U61'(mark'(X1), X2, X3) → mark'(U61'(X1, X2, X3))
U62'(mark'(X1), X2, X3) → mark'(U62'(X1, X2, X3))
U63'(mark'(X1), X2, X3) → mark'(U63'(X1, X2, X3))
U64'(mark'(X1), X2, X3) → mark'(U64'(X1, X2, X3))
s'(mark'(X)) → mark'(s'(X))
plus'(mark'(X1), X2) → mark'(plus'(X1, X2))
plus'(X1, mark'(X2)) → mark'(plus'(X1, X2))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(tt') → ok'(tt')
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(isNatKind'(X)) → isNatKind'(proper'(X))
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'(isNat'(X)) → isNat'(proper'(X))
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)) → U31'(proper'(X1), proper'(X2))
proper'(U32'(X)) → U32'(proper'(X))
proper'(U41'(X)) → U41'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X1, X2)) → U52'(proper'(X1), proper'(X2))
proper'(U61'(X1, X2, X3)) → U61'(proper'(X1), proper'(X2), proper'(X3))
proper'(U62'(X1, X2, X3)) → U62'(proper'(X1), proper'(X2), proper'(X3))
proper'(U63'(X1, X2, X3)) → U63'(proper'(X1), proper'(X2), proper'(X3))
proper'(U64'(X1, X2, X3)) → U64'(proper'(X1), proper'(X2), proper'(X3))
proper'(s'(X)) → s'(proper'(X))
proper'(plus'(X1, X2)) → plus'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
isNatKind'(ok'(X)) → ok'(isNatKind'(X))
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))
isNat'(ok'(X)) → ok'(isNat'(X))
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'(U31'(X1, X2))
U32'(ok'(X)) → ok'(U32'(X))
U41'(ok'(X)) → ok'(U41'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X1), ok'(X2)) → ok'(U52'(X1, X2))
U61'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U61'(X1, X2, X3))
U62'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U62'(X1, X2, X3))
U63'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U63'(X1, X2, X3))
U64'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U64'(X1, X2, X3))
s'(ok'(X)) → ok'(s'(X))
plus'(ok'(X1), ok'(X2)) → ok'(plus'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: 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'
tt' :: tt':mark':0':ok'
mark' :: 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'
isNatKind' :: 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'
isNat' :: 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'
U32' :: tt':mark':0':ok' → tt':mark':0':ok'
U41' :: tt':mark':0':ok' → tt':mark':0':ok'
U51' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
U52' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
U61' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
U62' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
U63' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
U64' :: 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'
plus' :: tt':mark':0':ok' → 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', U12', isNatKind', U13', U14', U15', isNat', U16', U22', U23', U32', U52', U62', U63', U64', s', plus', U11', U21', U31', U41', U51', U61', proper', top'

They will be analysed ascendingly in the following order:
U12' < active'
isNatKind' < active'
U13' < active'
U14' < active'
U15' < active'
isNat' < active'
U16' < active'
U22' < active'
U23' < active'
U32' < active'
U52' < active'
U62' < active'
U63' < active'
U64' < active'
s' < active'
plus' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
active' < top'
U12' < proper'
isNatKind' < proper'
U13' < proper'
U14' < proper'
U15' < proper'
isNat' < proper'
U16' < proper'
U22' < proper'
U23' < proper'
U32' < proper'
U52' < proper'
U62' < proper'
U63' < proper'
U64' < proper'
s' < proper'
plus' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < 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:
U12', active', isNatKind', U13', U14', U15', isNat', U16', U22', U23', U32', U52', U62', U63', U64', s', plus', U11', U21', U31', U41', U51', U61', proper', top'

Proved the following rewrite lemma:
U12'(_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:
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, +(_$n6, 1))), _gen_tt':mark':0':ok'3(_b1018), _gen_tt':mark':0':ok'3(_c1019)) →_R^Ω(1)
mark'(U12'(_gen_tt':mark':0':ok'3(+(1, _$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:
U12'(_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', U13', U14', U15', isNat', U16', U22', U23', U32', U52', U62', U63', U64', s', plus', U11', U21', U31', U41', U51', U61', proper', top'

They will be analysed ascendingly in the following order:
isNatKind' < active'
U13' < active'
U14' < active'
U15' < active'
isNat' < active'
U16' < active'
U22' < active'
U23' < active'
U32' < active'
U52' < active'
U62' < active'
U63' < active'
U64' < active'
s' < active'
plus' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
active' < top'
isNatKind' < proper'
U13' < proper'
U14' < proper'
U15' < proper'
isNat' < proper'
U16' < proper'
U22' < proper'
U23' < proper'
U32' < proper'
U52' < proper'
U62' < proper'
U63' < proper'
U64' < proper'
s' < proper'
plus' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
proper' < top'

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

Lemmas:
U12'(_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:
U13', active', U14', U15', isNat', U16', U22', U23', U32', U52', U62', U63', U64', s', plus', U11', U21', U31', U41', U51', U61', proper', top'

They will be analysed ascendingly in the following order:
U13' < active'
U14' < active'
U15' < active'
isNat' < active'
U16' < active'
U22' < active'
U23' < active'
U32' < active'
U52' < active'
U62' < active'
U63' < active'
U64' < active'
s' < active'
plus' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
active' < top'
U13' < proper'
U14' < proper'
U15' < proper'
isNat' < proper'
U16' < proper'
U22' < proper'
U23' < proper'
U32' < proper'
U52' < proper'
U62' < proper'
U63' < proper'
U64' < proper'
s' < proper'
plus' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
proper' < top'

Proved the following rewrite lemma:
U13'(_gen_tt':mark':0':ok'3(+(1, _n8894)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(_n8894)

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, +(_$n8895, 1))), _gen_tt':mark':0':ok'3(_b10555), _gen_tt':mark':0':ok'3(_c10556)) →_R^Ω(1)
mark'(U13'(_gen_tt':mark':0':ok'3(+(1, _$n8895)), _gen_tt':mark':0':ok'3(_b10555), _gen_tt':mark':0':ok'3(_c10556))) →_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:
U14', active', U15', isNat', U16', U22', U23', U32', U52', U62', U63', U64', s', plus', U11', U21', U31', U41', U51', U61', proper', top'

They will be analysed ascendingly in the following order:
U14' < active'
U15' < active'
isNat' < active'
U16' < active'
U22' < active'
U23' < active'
U32' < active'
U52' < active'
U62' < active'
U63' < active'
U64' < active'
s' < active'
plus' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
active' < top'
U14' < proper'
U15' < proper'
isNat' < proper'
U16' < proper'
U22' < proper'
U23' < proper'
U32' < proper'
U52' < proper'
U62' < proper'
U63' < proper'
U64' < proper'
s' < proper'
plus' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
proper' < top'

Proved the following rewrite lemma:
U14'(_gen_tt':mark':0':ok'3(+(1, _n18495)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(_n18495)

Induction Base:
U14'(_gen_tt':mark':0':ok'3(+(1, 0)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c))

Induction Step:
U14'(_gen_tt':mark':0':ok'3(+(1, +(_$n18496, 1))), _gen_tt':mark':0':ok'3(_b20804), _gen_tt':mark':0':ok'3(_c20805)) →_R^Ω(1)
mark'(U14'(_gen_tt':mark':0':ok'3(+(1, _$n18496)), _gen_tt':mark':0':ok'3(_b20804), _gen_tt':mark':0':ok'3(_c20805))) →_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:
U15', active', isNat', U16', U22', U23', U32', U52', U62', U63', U64', s', plus', U11', U21', U31', U41', U51', U61', proper', top'

They will be analysed ascendingly in the following order:
U15' < active'
isNat' < active'
U16' < active'
U22' < active'
U23' < active'
U32' < active'
U52' < active'
U62' < active'
U63' < active'
U64' < active'
s' < active'
plus' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
active' < top'
U15' < proper'
isNat' < proper'
U16' < proper'
U22' < proper'
U23' < proper'
U32' < proper'
U52' < proper'
U62' < proper'
U63' < proper'
U64' < proper'
s' < proper'
plus' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
proper' < top'

Proved the following rewrite lemma:
U15'(_gen_tt':mark':0':ok'3(+(1, _n28828)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(_n28828)

Induction Base:
U15'(_gen_tt':mark':0':ok'3(+(1, 0)), _gen_tt':mark':0':ok'3(b))

Induction Step:
U15'(_gen_tt':mark':0':ok'3(+(1, +(_$n28829, 1))), _gen_tt':mark':0':ok'3(_b30729)) →_R^Ω(1)
mark'(U15'(_gen_tt':mark':0':ok'3(+(1, _$n28829)), _gen_tt':mark':0':ok'3(_b30729))) →_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', U16', U22', U23', U32', U52', U62', U63', U64', s', plus', U11', U21', U31', U41', U51', U61', proper', top'

They will be analysed ascendingly in the following order:
isNat' < active'
U16' < active'
U22' < active'
U23' < active'
U32' < active'
U52' < active'
U62' < active'
U63' < active'
U64' < active'
s' < active'
plus' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
active' < top'
isNat' < proper'
U16' < proper'
U22' < proper'
U23' < proper'
U32' < proper'
U52' < proper'
U62' < proper'
U63' < proper'
U64' < proper'
s' < proper'
plus' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < 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:
U16', active', U22', U23', U32', U52', U62', U63', U64', s', plus', U11', U21', U31', U41', U51', U61', proper', top'

They will be analysed ascendingly in the following order:
U16' < active'
U22' < active'
U23' < active'
U32' < active'
U52' < active'
U62' < active'
U63' < active'
U64' < active'
s' < active'
plus' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
active' < top'
U16' < proper'
U22' < proper'
U23' < proper'
U32' < proper'
U52' < proper'
U62' < proper'
U63' < proper'
U64' < proper'
s' < proper'
plus' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
proper' < top'

Proved the following rewrite lemma:
U16'(_gen_tt':mark':0':ok'3(+(1, _n36848))) → _*4, rt ∈ Ω(_n36848)

Induction Base:
U16'(_gen_tt':mark':0':ok'3(+(1, 0)))

Induction Step:
U16'(_gen_tt':mark':0':ok'3(+(1, +(_$n36849, 1)))) →_R^Ω(1)
mark'(U16'(_gen_tt':mark':0':ok'3(+(1, _$n36849)))) →_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:
U22', active', U23', U32', U52', U62', U63', U64', s', plus', U11', U21', U31', U41', U51', U61', proper', top'

They will be analysed ascendingly in the following order:
U22' < active'
U23' < active'
U32' < active'
U52' < active'
U62' < active'
U63' < active'
U64' < active'
s' < active'
plus' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
active' < top'
U22' < proper'
U23' < proper'
U32' < proper'
U52' < proper'
U62' < proper'
U63' < proper'
U64' < proper'
s' < proper'
plus' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
proper' < top'

Proved the following rewrite lemma:
U22'(_gen_tt':mark':0':ok'3(+(1, _n42012)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(_n42012)

Induction Base:
U22'(_gen_tt':mark':0':ok'3(+(1, 0)), _gen_tt':mark':0':ok'3(b))

Induction Step:
U22'(_gen_tt':mark':0':ok'3(+(1, +(_$n42013, 1))), _gen_tt':mark':0':ok'3(_b44453)) →_R^Ω(1)
mark'(U22'(_gen_tt':mark':0':ok'3(+(1, _$n42013)), _gen_tt':mark':0':ok'3(_b44453))) →_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:
U23', active', U32', U52', U62', U63', U64', s', plus', U11', U21', U31', U41', U51', U61', proper', top'

They will be analysed ascendingly in the following order:
U23' < active'
U32' < active'
U52' < active'
U62' < active'
U63' < active'
U64' < active'
s' < active'
plus' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
active' < top'
U23' < proper'
U32' < proper'
U52' < proper'
U62' < proper'
U63' < proper'
U64' < proper'
s' < proper'
plus' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
proper' < top'

Proved the following rewrite lemma:
U23'(_gen_tt':mark':0':ok'3(+(1, _n50583))) → _*4, rt ∈ Ω(_n50583)

Induction Base:
U23'(_gen_tt':mark':0':ok'3(+(1, 0)))

Induction Step:
U23'(_gen_tt':mark':0':ok'3(+(1, +(_$n50584, 1)))) →_R^Ω(1)
mark'(U23'(_gen_tt':mark':0':ok'3(+(1, _$n50584)))) →_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:
U32', active', U52', U62', U63', U64', s', plus', U11', U21', U31', U41', U51', U61', proper', top'

They will be analysed ascendingly in the following order:
U32' < active'
U52' < active'
U62' < active'
U63' < active'
U64' < active'
s' < active'
plus' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
active' < top'
U32' < proper'
U52' < proper'
U62' < proper'
U63' < proper'
U64' < proper'
s' < proper'
plus' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
proper' < top'

Proved the following rewrite lemma:
U32'(_gen_tt':mark':0':ok'3(+(1, _n56062))) → _*4, rt ∈ Ω(_n56062)

Induction Base:
U32'(_gen_tt':mark':0':ok'3(+(1, 0)))

Induction Step:
U32'(_gen_tt':mark':0':ok'3(+(1, +(_$n56063, 1)))) →_R^Ω(1)
mark'(U32'(_gen_tt':mark':0':ok'3(+(1, _$n56063)))) →_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:
U52', active', U62', U63', U64', s', plus', U11', U21', U31', U41', U51', U61', proper', top'

They will be analysed ascendingly in the following order:
U52' < active'
U62' < active'
U63' < active'
U64' < active'
s' < active'
plus' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
active' < top'
U52' < proper'
U62' < proper'
U63' < proper'
U64' < proper'
s' < proper'
plus' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
proper' < top'

Proved the following rewrite lemma:
U52'(_gen_tt':mark':0':ok'3(+(1, _n61665)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(_n61665)

Induction Base:
U52'(_gen_tt':mark':0':ok'3(+(1, 0)), _gen_tt':mark':0':ok'3(b))

Induction Step:
U52'(_gen_tt':mark':0':ok'3(+(1, +(_$n61666, 1))), _gen_tt':mark':0':ok'3(_b64862)) →_R^Ω(1)
mark'(U52'(_gen_tt':mark':0':ok'3(+(1, _$n61666)), _gen_tt':mark':0':ok'3(_b64862))) →_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:
U62', active', U63', U64', s', plus', U11', U21', U31', U41', U51', U61', proper', top'

They will be analysed ascendingly in the following order:
U62' < active'
U63' < active'
U64' < active'
s' < active'
plus' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
active' < top'
U62' < proper'
U63' < proper'
U64' < proper'
s' < proper'
plus' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
proper' < top'

Proved the following rewrite lemma:
U62'(_gen_tt':mark':0':ok'3(+(1, _n71086)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(_n71086)

Induction Base:
U62'(_gen_tt':mark':0':ok'3(+(1, 0)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c))

Induction Step:
U62'(_gen_tt':mark':0':ok'3(+(1, +(_$n71087, 1))), _gen_tt':mark':0':ok'3(_b76473), _gen_tt':mark':0':ok'3(_c76474)) →_R^Ω(1)
mark'(U62'(_gen_tt':mark':0':ok'3(+(1, _$n71087)), _gen_tt':mark':0':ok'3(_b76473), _gen_tt':mark':0':ok'3(_c76474))) →_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:
U63', active', U64', s', plus', U11', U21', U31', U41', U51', U61', proper', top'

They will be analysed ascendingly in the following order:
U63' < active'
U64' < active'
s' < active'
plus' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
active' < top'
U63' < proper'
U64' < proper'
s' < proper'
plus' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
proper' < top'

Proved the following rewrite lemma:
U63'(_gen_tt':mark':0':ok'3(+(1, _n84860)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(_n84860)

Induction Base:
U63'(_gen_tt':mark':0':ok'3(+(1, 0)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c))

Induction Step:
U63'(_gen_tt':mark':0':ok'3(+(1, +(_$n84861, 1))), _gen_tt':mark':0':ok'3(_b90895), _gen_tt':mark':0':ok'3(_c90896)) →_R^Ω(1)
mark'(U63'(_gen_tt':mark':0':ok'3(+(1, _$n84861)), _gen_tt':mark':0':ok'3(_b90895), _gen_tt':mark':0':ok'3(_c90896))) →_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:
U64', active', s', plus', U11', U21', U31', U41', U51', U61', proper', top'

They will be analysed ascendingly in the following order:
U64' < active'
s' < active'
plus' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
active' < top'
U64' < proper'
s' < proper'
plus' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
proper' < top'

Proved the following rewrite lemma:
U64'(_gen_tt':mark':0':ok'3(+(1, _n99366)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(_n99366)

Induction Base:
U64'(_gen_tt':mark':0':ok'3(+(1, 0)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c))

Induction Step:
U64'(_gen_tt':mark':0':ok'3(+(1, +(_$n99367, 1))), _gen_tt':mark':0':ok'3(_b106049), _gen_tt':mark':0':ok'3(_c106050)) →_R^Ω(1)
mark'(U64'(_gen_tt':mark':0':ok'3(+(1, _$n99367)), _gen_tt':mark':0':ok'3(_b106049), _gen_tt':mark':0':ok'3(_c106050))) →_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:
s', active', plus', U11', U21', U31', U41', U51', U61', proper', top'

They will be analysed ascendingly in the following order:
s' < active'
plus' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
active' < top'
s' < proper'
plus' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
proper' < top'

Proved the following rewrite lemma:
s'(_gen_tt':mark':0':ok'3(+(1, _n114604))) → _*4, rt ∈ Ω(_n114604)

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

Induction Step:
s'(_gen_tt':mark':0':ok'3(+(1, +(_$n114605, 1)))) →_R^Ω(1)
mark'(s'(_gen_tt':mark':0':ok'3(+(1, _$n114605)))) →_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', U11', U21', U31', U41', U51', U61', proper', top'

They will be analysed ascendingly in the following order:
plus' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
active' < top'
plus' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
proper' < top'

Proved the following rewrite lemma:
plus'(_gen_tt':mark':0':ok'3(+(1, _n121296)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(_n121296)

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, +(_$n121297, 1))), _gen_tt':mark':0':ok'3(_b126545)) →_R^Ω(1)
mark'(plus'(_gen_tt':mark':0':ok'3(+(1, _$n121297)), _gen_tt':mark':0':ok'3(_b126545))) →_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:
U11', active', U21', U31', U41', U51', U61', proper', top'

They will be analysed ascendingly in the following order:
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
active' < top'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
proper' < top'

Proved the following rewrite lemma:
U11'(_gen_tt':mark':0':ok'3(+(1, _n133030)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(_n133030)

Induction Base:
U11'(_gen_tt':mark':0':ok'3(+(1, 0)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c))

Induction Step:
U11'(_gen_tt':mark':0':ok'3(+(1, +(_$n133031, 1))), _gen_tt':mark':0':ok'3(_b141171), _gen_tt':mark':0':ok'3(_c141172)) →_R^Ω(1)
mark'(U11'(_gen_tt':mark':0':ok'3(+(1, _$n133031)), _gen_tt':mark':0':ok'3(_b141171), _gen_tt':mark':0':ok'3(_c141172))) →_IH
mark'(_*4)

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

Lemmas:
U12'(_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)
U13'(_gen_tt':mark':0':ok'3(+(1, _n8894)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(_n8894)
U14'(_gen_tt':mark':0':ok'3(+(1, _n18495)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(_n18495)
U15'(_gen_tt':mark':0':ok'3(+(1, _n28828)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(_n28828)
U16'(_gen_tt':mark':0':ok'3(+(1, _n36848))) → _*4, rt ∈ Ω(_n36848)
U22'(_gen_tt':mark':0':ok'3(+(1, _n42012)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(_n42012)
U23'(_gen_tt':mark':0':ok'3(+(1, _n50583))) → _*4, rt ∈ Ω(_n50583)
U32'(_gen_tt':mark':0':ok'3(+(1, _n56062))) → _*4, rt ∈ Ω(_n56062)
U52'(_gen_tt':mark':0':ok'3(+(1, _n61665)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(_n61665)
U62'(_gen_tt':mark':0':ok'3(+(1, _n71086)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(_n71086)
U63'(_gen_tt':mark':0':ok'3(+(1, _n84860)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(_n84860)
U64'(_gen_tt':mark':0':ok'3(+(1, _n99366)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(_n99366)
s'(_gen_tt':mark':0':ok'3(+(1, _n114604))) → _*4, rt ∈ Ω(_n114604)
plus'(_gen_tt':mark':0':ok'3(+(1, _n121296)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(_n121296)
U11'(_gen_tt':mark':0':ok'3(+(1, _n133030)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(_n133030)

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:
U21', active', U31', U41', U51', U61', proper', top'

They will be analysed ascendingly in the following order:
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
active' < top'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
proper' < top'

Proved the following rewrite lemma:
U21'(_gen_tt':mark':0':ok'3(+(1, _n149903)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(_n149903)

Induction Base:
U21'(_gen_tt':mark':0':ok'3(+(1, 0)), _gen_tt':mark':0':ok'3(b))

Induction Step:
U21'(_gen_tt':mark':0':ok'3(+(1, +(_$n149904, 1))), _gen_tt':mark':0':ok'3(_b155692)) →_R^Ω(1)
mark'(U21'(_gen_tt':mark':0':ok'3(+(1, _$n149904)), _gen_tt':mark':0':ok'3(_b155692))) →_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:
U31', active', U41', U51', U61', proper', top'

They will be analysed ascendingly in the following order:
U31' < active'
U41' < active'
U51' < active'
U61' < active'
active' < top'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
proper' < top'

Proved the following rewrite lemma:
U31'(_gen_tt':mark':0':ok'3(+(1, _n162281)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(_n162281)

Induction Base:
U31'(_gen_tt':mark':0':ok'3(+(1, 0)), _gen_tt':mark':0':ok'3(b))

Induction Step:
U31'(_gen_tt':mark':0':ok'3(+(1, +(_$n162282, 1))), _gen_tt':mark':0':ok'3(_b168394)) →_R^Ω(1)
mark'(U31'(_gen_tt':mark':0':ok'3(+(1, _$n162282)), _gen_tt':mark':0':ok'3(_b168394))) →_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:
U41', active', U51', U61', proper', top'

They will be analysed ascendingly in the following order:
U41' < active'
U51' < active'
U61' < active'
active' < top'
U41' < proper'
U51' < proper'
U61' < proper'
proper' < top'

Proved the following rewrite lemma:
U41'(_gen_tt':mark':0':ok'3(+(1, _n175027))) → _*4, rt ∈ Ω(_n175027)

Induction Base:
U41'(_gen_tt':mark':0':ok'3(+(1, 0)))

Induction Step:
U41'(_gen_tt':mark':0':ok'3(+(1, +(_$n175028, 1)))) →_R^Ω(1)
mark'(U41'(_gen_tt':mark':0':ok'3(+(1, _$n175028)))) →_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:
U51', active', U61', proper', top'

They will be analysed ascendingly in the following order:
U51' < active'
U61' < active'
active' < top'
U51' < proper'
U61' < proper'
proper' < top'

Proved the following rewrite lemma:
U51'(_gen_tt':mark':0':ok'3(+(1, _n182674)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(_n182674)

Induction Base:
U51'(_gen_tt':mark':0':ok'3(+(1, 0)), _gen_tt':mark':0':ok'3(b))

Induction Step:
U51'(_gen_tt':mark':0':ok'3(+(1, +(_$n182675, 1))), _gen_tt':mark':0':ok'3(_b189327)) →_R^Ω(1)
mark'(U51'(_gen_tt':mark':0':ok'3(+(1, _$n182675)), _gen_tt':mark':0':ok'3(_b189327))) →_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:
U61', active', proper', top'

They will be analysed ascendingly in the following order:
U61' < active'
active' < top'
U61' < proper'
proper' < top'

Proved the following rewrite lemma:
U61'(_gen_tt':mark':0':ok'3(+(1, _n196029)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(_n196029)

Induction Base:
U61'(_gen_tt':mark':0':ok'3(+(1, 0)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c))

Induction Step:
U61'(_gen_tt':mark':0':ok'3(+(1, +(_$n196030, 1))), _gen_tt':mark':0':ok'3(_b206600), _gen_tt':mark':0':ok'3(_c206601)) →_R^Ω(1)
mark'(U61'(_gen_tt':mark':0':ok'3(+(1, _$n196030)), _gen_tt':mark':0':ok'3(_b206600), _gen_tt':mark':0':ok'3(_c206601))) →_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:
active', proper', top'

They will be analysed ascendingly in the following order:
active' < top'
proper' < top'