WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) CpxTRS
(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 4 ms]
(2) CpxTRS
    (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
    (4) CpxTRS
    (5) CpxTrsMatchBoundsTAProof [FINISHED, 895 ms]
    (6) BOUNDS(1, n^1)
(7) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CpxTRS
    (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) typed CpxTrs
    (11) OrderProof [LOWER BOUND(ID), 8 ms]
    (12) typed CpxTrs
    (13) RewriteLemmaProof [LOWER BOUND(ID), 443 ms]
    (14) BEST
        (15) proven lower bound
            (16) LowerBoundPropagationProof [FINISHED, 0 ms]
            (17) BOUNDS(n^1, INF)
        (18) typed CpxTrs
            (19) RewriteLemmaProof [LOWER BOUND(ID), 149 ms]
            (20) typed CpxTrs
            (21) RewriteLemmaProof [LOWER BOUND(ID), 125 ms]
            (22) typed CpxTrs
            (23) RewriteLemmaProof [LOWER BOUND(ID), 107 ms]
            (24) typed CpxTrs
            (25) RewriteLemmaProof [LOWER BOUND(ID), 78 ms]
            (26) typed CpxTrs
            (27) RewriteLemmaProof [LOWER BOUND(ID), 106 ms]
            (28) typed CpxTrs
            (29) RewriteLemmaProof [LOWER BOUND(ID), 103 ms]
            (30) typed CpxTrs
            (31) RewriteLemmaProof [LOWER BOUND(ID), 92 ms]
            (32) typed CpxTrs
            (33) RewriteLemmaProof [LOWER BOUND(ID), 132 ms]
            (34) typed CpxTrs
            (35) RewriteLemmaProof [LOWER BOUND(ID), 222 ms]
            (36) typed CpxTrs
            (37) RewriteLemmaProof [LOWER BOUND(ID), 150 ms]
            (38) typed CpxTrs
            (39) RewriteLemmaProof [LOWER BOUND(ID), 169 ms]
            (40) typed CpxTrs
            (41) RewriteLemmaProof [LOWER BOUND(ID), 92 ms]
            (42) typed CpxTrs
            (43) RewriteLemmaProof [LOWER BOUND(ID), 61 ms]
            (44) typed CpxTrs
            (45) RewriteLemmaProof [LOWER BOUND(ID), 160 ms]
            (46) typed CpxTrs
            (47) RewriteLemmaProof [LOWER BOUND(ID), 116 ms]
            (48) typed CpxTrs
            (49) RewriteLemmaProof [LOWER BOUND(ID), 112 ms]
            (50) typed CpxTrs
            (51) RewriteLemmaProof [LOWER BOUND(ID), 45 ms]
            (52) typed CpxTrs
            (53) RewriteLemmaProof [LOWER BOUND(ID), 113 ms]
            (54) typed CpxTrs
            (55) RewriteLemmaProof [LOWER BOUND(ID), 219 ms]
            (56) typed CpxTrs


----------------------------------------

(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).


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))

S is empty.
Rewrite Strategy: FULL
----------------------------------------

(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
The following defined symbols can occur below the 0th argument of top: proper, active
The following defined symbols can occur below the 0th argument of proper: proper, active
The following defined symbols can occur below the 0th argument of active: proper, active

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
active(U11(tt, V1, V2)) -> mark(U12(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))
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(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))

----------------------------------------

(2)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   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(tt) -> ok(tt)
   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))

S is empty.
Rewrite Strategy: FULL
----------------------------------------

(3) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(4)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   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(tt) -> ok(tt)
   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))

S is empty.
Rewrite Strategy: FULL
----------------------------------------

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

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]
transitions: 
mark0(0) -> 0
tt0() -> 0
ok0(0) -> 0
00() -> 0
active0(0) -> 0
U110(0, 0, 0) -> 1
U120(0, 0, 0) -> 2
U130(0, 0, 0) -> 3
U140(0, 0, 0) -> 4
U150(0, 0) -> 5
U160(0) -> 6
U210(0, 0) -> 7
U220(0, 0) -> 8
U230(0) -> 9
U310(0, 0) -> 10
U320(0) -> 11
U410(0) -> 12
U510(0, 0) -> 13
U520(0, 0) -> 14
U610(0, 0, 0) -> 15
U620(0, 0, 0) -> 16
U630(0, 0, 0) -> 17
U640(0, 0, 0) -> 18
s0(0) -> 19
plus0(0, 0) -> 20
proper0(0) -> 21
isNatKind0(0) -> 22
isNat0(0) -> 23
top0(0) -> 24
U111(0, 0, 0) -> 25
mark1(25) -> 1
U121(0, 0, 0) -> 26
mark1(26) -> 2
U131(0, 0, 0) -> 27
mark1(27) -> 3
U141(0, 0, 0) -> 28
mark1(28) -> 4
U151(0, 0) -> 29
mark1(29) -> 5
U161(0) -> 30
mark1(30) -> 6
U211(0, 0) -> 31
mark1(31) -> 7
U221(0, 0) -> 32
mark1(32) -> 8
U231(0) -> 33
mark1(33) -> 9
U311(0, 0) -> 34
mark1(34) -> 10
U321(0) -> 35
mark1(35) -> 11
U411(0) -> 36
mark1(36) -> 12
U511(0, 0) -> 37
mark1(37) -> 13
U521(0, 0) -> 38
mark1(38) -> 14
U611(0, 0, 0) -> 39
mark1(39) -> 15
U621(0, 0, 0) -> 40
mark1(40) -> 16
U631(0, 0, 0) -> 41
mark1(41) -> 17
U641(0, 0, 0) -> 42
mark1(42) -> 18
s1(0) -> 43
mark1(43) -> 19
plus1(0, 0) -> 44
mark1(44) -> 20
tt1() -> 45
ok1(45) -> 21
01() -> 46
ok1(46) -> 21
U111(0, 0, 0) -> 47
ok1(47) -> 1
U121(0, 0, 0) -> 48
ok1(48) -> 2
isNatKind1(0) -> 49
ok1(49) -> 22
U131(0, 0, 0) -> 50
ok1(50) -> 3
U141(0, 0, 0) -> 51
ok1(51) -> 4
U151(0, 0) -> 52
ok1(52) -> 5
isNat1(0) -> 53
ok1(53) -> 23
U161(0) -> 54
ok1(54) -> 6
U211(0, 0) -> 55
ok1(55) -> 7
U221(0, 0) -> 56
ok1(56) -> 8
U231(0) -> 57
ok1(57) -> 9
U311(0, 0) -> 58
ok1(58) -> 10
U321(0) -> 59
ok1(59) -> 11
U411(0) -> 60
ok1(60) -> 12
U511(0, 0) -> 61
ok1(61) -> 13
U521(0, 0) -> 62
ok1(62) -> 14
U611(0, 0, 0) -> 63
ok1(63) -> 15
U621(0, 0, 0) -> 64
ok1(64) -> 16
U631(0, 0, 0) -> 65
ok1(65) -> 17
U641(0, 0, 0) -> 66
ok1(66) -> 18
s1(0) -> 67
ok1(67) -> 19
plus1(0, 0) -> 68
ok1(68) -> 20
proper1(0) -> 69
top1(69) -> 24
active1(0) -> 70
top1(70) -> 24
mark1(25) -> 25
mark1(25) -> 47
mark1(26) -> 26
mark1(26) -> 48
mark1(27) -> 27
mark1(27) -> 50
mark1(28) -> 28
mark1(28) -> 51
mark1(29) -> 29
mark1(29) -> 52
mark1(30) -> 30
mark1(30) -> 54
mark1(31) -> 31
mark1(31) -> 55
mark1(32) -> 32
mark1(32) -> 56
mark1(33) -> 33
mark1(33) -> 57
mark1(34) -> 34
mark1(34) -> 58
mark1(35) -> 35
mark1(35) -> 59
mark1(36) -> 36
mark1(36) -> 60
mark1(37) -> 37
mark1(37) -> 61
mark1(38) -> 38
mark1(38) -> 62
mark1(39) -> 39
mark1(39) -> 63
mark1(40) -> 40
mark1(40) -> 64
mark1(41) -> 41
mark1(41) -> 65
mark1(42) -> 42
mark1(42) -> 66
mark1(43) -> 43
mark1(43) -> 67
mark1(44) -> 44
mark1(44) -> 68
ok1(45) -> 69
ok1(46) -> 69
ok1(47) -> 25
ok1(47) -> 47
ok1(48) -> 26
ok1(48) -> 48
ok1(49) -> 49
ok1(50) -> 27
ok1(50) -> 50
ok1(51) -> 28
ok1(51) -> 51
ok1(52) -> 29
ok1(52) -> 52
ok1(53) -> 53
ok1(54) -> 30
ok1(54) -> 54
ok1(55) -> 31
ok1(55) -> 55
ok1(56) -> 32
ok1(56) -> 56
ok1(57) -> 33
ok1(57) -> 57
ok1(58) -> 34
ok1(58) -> 58
ok1(59) -> 35
ok1(59) -> 59
ok1(60) -> 36
ok1(60) -> 60
ok1(61) -> 37
ok1(61) -> 61
ok1(62) -> 38
ok1(62) -> 62
ok1(63) -> 39
ok1(63) -> 63
ok1(64) -> 40
ok1(64) -> 64
ok1(65) -> 41
ok1(65) -> 65
ok1(66) -> 42
ok1(66) -> 66
ok1(67) -> 43
ok1(67) -> 67
ok1(68) -> 44
ok1(68) -> 68
active2(45) -> 71
top2(71) -> 24
active2(46) -> 71

----------------------------------------

(6)
BOUNDS(1, n^1)

----------------------------------------

(7) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(8)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


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))

S is empty.
Rewrite Strategy: FULL
----------------------------------------

(9) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(10)
Obligation:
TRS:
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':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok

----------------------------------------

(11) OrderProof (LOWER BOUND(ID))
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

----------------------------------------

(12)
Obligation:
TRS:
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':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok


Generator Equations:
gen_tt:mark:0':ok3_0(0) <=> tt
gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(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

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

----------------------------------------

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0)

Induction Base:
U12(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))

Induction Step:
U12(gen_tt:mark:0':ok3_0(+(1, +(n5_0, 1))), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) ->_R^Omega(1)
mark(U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(14)
Complex Obligation (BEST)

----------------------------------------

(15)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
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':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok


Generator Equations:
gen_tt:mark:0':ok3_0(0) <=> tt
gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(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

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

----------------------------------------

(16) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(17)
BOUNDS(n^1, INF)

----------------------------------------

(18)
Obligation:
TRS:
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':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok


Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0)


Generator Equations:
gen_tt:mark:0':ok3_0(0) <=> tt
gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(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

----------------------------------------

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U13(gen_tt:mark:0':ok3_0(+(1, n3519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n3519_0)

Induction Base:
U13(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))

Induction Step:
U13(gen_tt:mark:0':ok3_0(+(1, +(n3519_0, 1))), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) ->_R^Omega(1)
mark(U13(gen_tt:mark:0':ok3_0(+(1, n3519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(20)
Obligation:
TRS:
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':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok


Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0)
U13(gen_tt:mark:0':ok3_0(+(1, n3519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n3519_0)


Generator Equations:
gen_tt:mark:0':ok3_0(0) <=> tt
gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(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

----------------------------------------

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U14(gen_tt:mark:0':ok3_0(+(1, n7631_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n7631_0)

Induction Base:
U14(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))

Induction Step:
U14(gen_tt:mark:0':ok3_0(+(1, +(n7631_0, 1))), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) ->_R^Omega(1)
mark(U14(gen_tt:mark:0':ok3_0(+(1, n7631_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(22)
Obligation:
TRS:
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':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok


Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0)
U13(gen_tt:mark:0':ok3_0(+(1, n3519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n3519_0)
U14(gen_tt:mark:0':ok3_0(+(1, n7631_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n7631_0)


Generator Equations:
gen_tt:mark:0':ok3_0(0) <=> tt
gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(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

----------------------------------------

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U15(gen_tt:mark:0':ok3_0(+(1, n12352_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n12352_0)

Induction Base:
U15(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b))

Induction Step:
U15(gen_tt:mark:0':ok3_0(+(1, +(n12352_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1)
mark(U15(gen_tt:mark:0':ok3_0(+(1, n12352_0)), gen_tt:mark:0':ok3_0(b))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(24)
Obligation:
TRS:
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':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok


Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0)
U13(gen_tt:mark:0':ok3_0(+(1, n3519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n3519_0)
U14(gen_tt:mark:0':ok3_0(+(1, n7631_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n7631_0)
U15(gen_tt:mark:0':ok3_0(+(1, n12352_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n12352_0)


Generator Equations:
gen_tt:mark:0':ok3_0(0) <=> tt
gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(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

----------------------------------------

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U16(gen_tt:mark:0':ok3_0(+(1, n15653_0))) -> *4_0, rt in Omega(n15653_0)

Induction Base:
U16(gen_tt:mark:0':ok3_0(+(1, 0)))

Induction Step:
U16(gen_tt:mark:0':ok3_0(+(1, +(n15653_0, 1)))) ->_R^Omega(1)
mark(U16(gen_tt:mark:0':ok3_0(+(1, n15653_0)))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(26)
Obligation:
TRS:
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':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok


Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0)
U13(gen_tt:mark:0':ok3_0(+(1, n3519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n3519_0)
U14(gen_tt:mark:0':ok3_0(+(1, n7631_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n7631_0)
U15(gen_tt:mark:0':ok3_0(+(1, n12352_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n12352_0)
U16(gen_tt:mark:0':ok3_0(+(1, n15653_0))) -> *4_0, rt in Omega(n15653_0)


Generator Equations:
gen_tt:mark:0':ok3_0(0) <=> tt
gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(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

----------------------------------------

(27) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U22(gen_tt:mark:0':ok3_0(+(1, n17015_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n17015_0)

Induction Base:
U22(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b))

Induction Step:
U22(gen_tt:mark:0':ok3_0(+(1, +(n17015_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1)
mark(U22(gen_tt:mark:0':ok3_0(+(1, n17015_0)), gen_tt:mark:0':ok3_0(b))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(28)
Obligation:
TRS:
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':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok


Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0)
U13(gen_tt:mark:0':ok3_0(+(1, n3519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n3519_0)
U14(gen_tt:mark:0':ok3_0(+(1, n7631_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n7631_0)
U15(gen_tt:mark:0':ok3_0(+(1, n12352_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n12352_0)
U16(gen_tt:mark:0':ok3_0(+(1, n15653_0))) -> *4_0, rt in Omega(n15653_0)
U22(gen_tt:mark:0':ok3_0(+(1, n17015_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n17015_0)


Generator Equations:
gen_tt:mark:0':ok3_0(0) <=> tt
gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(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

----------------------------------------

(29) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U23(gen_tt:mark:0':ok3_0(+(1, n20800_0))) -> *4_0, rt in Omega(n20800_0)

Induction Base:
U23(gen_tt:mark:0':ok3_0(+(1, 0)))

Induction Step:
U23(gen_tt:mark:0':ok3_0(+(1, +(n20800_0, 1)))) ->_R^Omega(1)
mark(U23(gen_tt:mark:0':ok3_0(+(1, n20800_0)))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(30)
Obligation:
TRS:
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':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok


Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0)
U13(gen_tt:mark:0':ok3_0(+(1, n3519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n3519_0)
U14(gen_tt:mark:0':ok3_0(+(1, n7631_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n7631_0)
U15(gen_tt:mark:0':ok3_0(+(1, n12352_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n12352_0)
U16(gen_tt:mark:0':ok3_0(+(1, n15653_0))) -> *4_0, rt in Omega(n15653_0)
U22(gen_tt:mark:0':ok3_0(+(1, n17015_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n17015_0)
U23(gen_tt:mark:0':ok3_0(+(1, n20800_0))) -> *4_0, rt in Omega(n20800_0)


Generator Equations:
gen_tt:mark:0':ok3_0(0) <=> tt
gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(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

----------------------------------------

(31) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U32(gen_tt:mark:0':ok3_0(+(1, n22413_0))) -> *4_0, rt in Omega(n22413_0)

Induction Base:
U32(gen_tt:mark:0':ok3_0(+(1, 0)))

Induction Step:
U32(gen_tt:mark:0':ok3_0(+(1, +(n22413_0, 1)))) ->_R^Omega(1)
mark(U32(gen_tt:mark:0':ok3_0(+(1, n22413_0)))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(32)
Obligation:
TRS:
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':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok


Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0)
U13(gen_tt:mark:0':ok3_0(+(1, n3519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n3519_0)
U14(gen_tt:mark:0':ok3_0(+(1, n7631_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n7631_0)
U15(gen_tt:mark:0':ok3_0(+(1, n12352_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n12352_0)
U16(gen_tt:mark:0':ok3_0(+(1, n15653_0))) -> *4_0, rt in Omega(n15653_0)
U22(gen_tt:mark:0':ok3_0(+(1, n17015_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n17015_0)
U23(gen_tt:mark:0':ok3_0(+(1, n20800_0))) -> *4_0, rt in Omega(n20800_0)
U32(gen_tt:mark:0':ok3_0(+(1, n22413_0))) -> *4_0, rt in Omega(n22413_0)


Generator Equations:
gen_tt:mark:0':ok3_0(0) <=> tt
gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(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

----------------------------------------

(33) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U52(gen_tt:mark:0':ok3_0(+(1, n24127_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n24127_0)

Induction Base:
U52(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b))

Induction Step:
U52(gen_tt:mark:0':ok3_0(+(1, +(n24127_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1)
mark(U52(gen_tt:mark:0':ok3_0(+(1, n24127_0)), gen_tt:mark:0':ok3_0(b))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(34)
Obligation:
TRS:
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':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok


Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0)
U13(gen_tt:mark:0':ok3_0(+(1, n3519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n3519_0)
U14(gen_tt:mark:0':ok3_0(+(1, n7631_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n7631_0)
U15(gen_tt:mark:0':ok3_0(+(1, n12352_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n12352_0)
U16(gen_tt:mark:0':ok3_0(+(1, n15653_0))) -> *4_0, rt in Omega(n15653_0)
U22(gen_tt:mark:0':ok3_0(+(1, n17015_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n17015_0)
U23(gen_tt:mark:0':ok3_0(+(1, n20800_0))) -> *4_0, rt in Omega(n20800_0)
U32(gen_tt:mark:0':ok3_0(+(1, n22413_0))) -> *4_0, rt in Omega(n22413_0)
U52(gen_tt:mark:0':ok3_0(+(1, n24127_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n24127_0)


Generator Equations:
gen_tt:mark:0':ok3_0(0) <=> tt
gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(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

----------------------------------------

(35) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U62(gen_tt:mark:0':ok3_0(+(1, n28634_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n28634_0)

Induction Base:
U62(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))

Induction Step:
U62(gen_tt:mark:0':ok3_0(+(1, +(n28634_0, 1))), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) ->_R^Omega(1)
mark(U62(gen_tt:mark:0':ok3_0(+(1, n28634_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(36)
Obligation:
TRS:
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':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok


Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0)
U13(gen_tt:mark:0':ok3_0(+(1, n3519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n3519_0)
U14(gen_tt:mark:0':ok3_0(+(1, n7631_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n7631_0)
U15(gen_tt:mark:0':ok3_0(+(1, n12352_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n12352_0)
U16(gen_tt:mark:0':ok3_0(+(1, n15653_0))) -> *4_0, rt in Omega(n15653_0)
U22(gen_tt:mark:0':ok3_0(+(1, n17015_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n17015_0)
U23(gen_tt:mark:0':ok3_0(+(1, n20800_0))) -> *4_0, rt in Omega(n20800_0)
U32(gen_tt:mark:0':ok3_0(+(1, n22413_0))) -> *4_0, rt in Omega(n22413_0)
U52(gen_tt:mark:0':ok3_0(+(1, n24127_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n24127_0)
U62(gen_tt:mark:0':ok3_0(+(1, n28634_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n28634_0)


Generator Equations:
gen_tt:mark:0':ok3_0(0) <=> tt
gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(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

----------------------------------------

(37) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U63(gen_tt:mark:0':ok3_0(+(1, n36295_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n36295_0)

Induction Base:
U63(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))

Induction Step:
U63(gen_tt:mark:0':ok3_0(+(1, +(n36295_0, 1))), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) ->_R^Omega(1)
mark(U63(gen_tt:mark:0':ok3_0(+(1, n36295_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(38)
Obligation:
TRS:
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':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok


Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0)
U13(gen_tt:mark:0':ok3_0(+(1, n3519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n3519_0)
U14(gen_tt:mark:0':ok3_0(+(1, n7631_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n7631_0)
U15(gen_tt:mark:0':ok3_0(+(1, n12352_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n12352_0)
U16(gen_tt:mark:0':ok3_0(+(1, n15653_0))) -> *4_0, rt in Omega(n15653_0)
U22(gen_tt:mark:0':ok3_0(+(1, n17015_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n17015_0)
U23(gen_tt:mark:0':ok3_0(+(1, n20800_0))) -> *4_0, rt in Omega(n20800_0)
U32(gen_tt:mark:0':ok3_0(+(1, n22413_0))) -> *4_0, rt in Omega(n22413_0)
U52(gen_tt:mark:0':ok3_0(+(1, n24127_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n24127_0)
U62(gen_tt:mark:0':ok3_0(+(1, n28634_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n28634_0)
U63(gen_tt:mark:0':ok3_0(+(1, n36295_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n36295_0)


Generator Equations:
gen_tt:mark:0':ok3_0(0) <=> tt
gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(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

----------------------------------------

(39) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U64(gen_tt:mark:0':ok3_0(+(1, n44565_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n44565_0)

Induction Base:
U64(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))

Induction Step:
U64(gen_tt:mark:0':ok3_0(+(1, +(n44565_0, 1))), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) ->_R^Omega(1)
mark(U64(gen_tt:mark:0':ok3_0(+(1, n44565_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(40)
Obligation:
TRS:
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':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok


Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0)
U13(gen_tt:mark:0':ok3_0(+(1, n3519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n3519_0)
U14(gen_tt:mark:0':ok3_0(+(1, n7631_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n7631_0)
U15(gen_tt:mark:0':ok3_0(+(1, n12352_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n12352_0)
U16(gen_tt:mark:0':ok3_0(+(1, n15653_0))) -> *4_0, rt in Omega(n15653_0)
U22(gen_tt:mark:0':ok3_0(+(1, n17015_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n17015_0)
U23(gen_tt:mark:0':ok3_0(+(1, n20800_0))) -> *4_0, rt in Omega(n20800_0)
U32(gen_tt:mark:0':ok3_0(+(1, n22413_0))) -> *4_0, rt in Omega(n22413_0)
U52(gen_tt:mark:0':ok3_0(+(1, n24127_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n24127_0)
U62(gen_tt:mark:0':ok3_0(+(1, n28634_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n28634_0)
U63(gen_tt:mark:0':ok3_0(+(1, n36295_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n36295_0)
U64(gen_tt:mark:0':ok3_0(+(1, n44565_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n44565_0)


Generator Equations:
gen_tt:mark:0':ok3_0(0) <=> tt
gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(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

----------------------------------------

(41) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
s(gen_tt:mark:0':ok3_0(+(1, n53444_0))) -> *4_0, rt in Omega(n53444_0)

Induction Base:
s(gen_tt:mark:0':ok3_0(+(1, 0)))

Induction Step:
s(gen_tt:mark:0':ok3_0(+(1, +(n53444_0, 1)))) ->_R^Omega(1)
mark(s(gen_tt:mark:0':ok3_0(+(1, n53444_0)))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(42)
Obligation:
TRS:
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':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok


Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0)
U13(gen_tt:mark:0':ok3_0(+(1, n3519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n3519_0)
U14(gen_tt:mark:0':ok3_0(+(1, n7631_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n7631_0)
U15(gen_tt:mark:0':ok3_0(+(1, n12352_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n12352_0)
U16(gen_tt:mark:0':ok3_0(+(1, n15653_0))) -> *4_0, rt in Omega(n15653_0)
U22(gen_tt:mark:0':ok3_0(+(1, n17015_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n17015_0)
U23(gen_tt:mark:0':ok3_0(+(1, n20800_0))) -> *4_0, rt in Omega(n20800_0)
U32(gen_tt:mark:0':ok3_0(+(1, n22413_0))) -> *4_0, rt in Omega(n22413_0)
U52(gen_tt:mark:0':ok3_0(+(1, n24127_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n24127_0)
U62(gen_tt:mark:0':ok3_0(+(1, n28634_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n28634_0)
U63(gen_tt:mark:0':ok3_0(+(1, n36295_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n36295_0)
U64(gen_tt:mark:0':ok3_0(+(1, n44565_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n44565_0)
s(gen_tt:mark:0':ok3_0(+(1, n53444_0))) -> *4_0, rt in Omega(n53444_0)


Generator Equations:
gen_tt:mark:0':ok3_0(0) <=> tt
gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(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

----------------------------------------

(43) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
plus(gen_tt:mark:0':ok3_0(+(1, n56006_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n56006_0)

Induction Base:
plus(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b))

Induction Step:
plus(gen_tt:mark:0':ok3_0(+(1, +(n56006_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1)
mark(plus(gen_tt:mark:0':ok3_0(+(1, n56006_0)), gen_tt:mark:0':ok3_0(b))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(44)
Obligation:
TRS:
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':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok


Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0)
U13(gen_tt:mark:0':ok3_0(+(1, n3519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n3519_0)
U14(gen_tt:mark:0':ok3_0(+(1, n7631_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n7631_0)
U15(gen_tt:mark:0':ok3_0(+(1, n12352_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n12352_0)
U16(gen_tt:mark:0':ok3_0(+(1, n15653_0))) -> *4_0, rt in Omega(n15653_0)
U22(gen_tt:mark:0':ok3_0(+(1, n17015_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n17015_0)
U23(gen_tt:mark:0':ok3_0(+(1, n20800_0))) -> *4_0, rt in Omega(n20800_0)
U32(gen_tt:mark:0':ok3_0(+(1, n22413_0))) -> *4_0, rt in Omega(n22413_0)
U52(gen_tt:mark:0':ok3_0(+(1, n24127_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n24127_0)
U62(gen_tt:mark:0':ok3_0(+(1, n28634_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n28634_0)
U63(gen_tt:mark:0':ok3_0(+(1, n36295_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n36295_0)
U64(gen_tt:mark:0':ok3_0(+(1, n44565_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n44565_0)
s(gen_tt:mark:0':ok3_0(+(1, n53444_0))) -> *4_0, rt in Omega(n53444_0)
plus(gen_tt:mark:0':ok3_0(+(1, n56006_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n56006_0)


Generator Equations:
gen_tt:mark:0':ok3_0(0) <=> tt
gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(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

----------------------------------------

(45) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U11(gen_tt:mark:0':ok3_0(+(1, n62438_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n62438_0)

Induction Base:
U11(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))

Induction Step:
U11(gen_tt:mark:0':ok3_0(+(1, +(n62438_0, 1))), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) ->_R^Omega(1)
mark(U11(gen_tt:mark:0':ok3_0(+(1, n62438_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(46)
Obligation:
TRS:
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':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok


Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0)
U13(gen_tt:mark:0':ok3_0(+(1, n3519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n3519_0)
U14(gen_tt:mark:0':ok3_0(+(1, n7631_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n7631_0)
U15(gen_tt:mark:0':ok3_0(+(1, n12352_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n12352_0)
U16(gen_tt:mark:0':ok3_0(+(1, n15653_0))) -> *4_0, rt in Omega(n15653_0)
U22(gen_tt:mark:0':ok3_0(+(1, n17015_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n17015_0)
U23(gen_tt:mark:0':ok3_0(+(1, n20800_0))) -> *4_0, rt in Omega(n20800_0)
U32(gen_tt:mark:0':ok3_0(+(1, n22413_0))) -> *4_0, rt in Omega(n22413_0)
U52(gen_tt:mark:0':ok3_0(+(1, n24127_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n24127_0)
U62(gen_tt:mark:0':ok3_0(+(1, n28634_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n28634_0)
U63(gen_tt:mark:0':ok3_0(+(1, n36295_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n36295_0)
U64(gen_tt:mark:0':ok3_0(+(1, n44565_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n44565_0)
s(gen_tt:mark:0':ok3_0(+(1, n53444_0))) -> *4_0, rt in Omega(n53444_0)
plus(gen_tt:mark:0':ok3_0(+(1, n56006_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n56006_0)
U11(gen_tt:mark:0':ok3_0(+(1, n62438_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n62438_0)


Generator Equations:
gen_tt:mark:0':ok3_0(0) <=> tt
gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(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

----------------------------------------

(47) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U21(gen_tt:mark:0':ok3_0(+(1, n72703_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n72703_0)

Induction Base:
U21(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b))

Induction Step:
U21(gen_tt:mark:0':ok3_0(+(1, +(n72703_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1)
mark(U21(gen_tt:mark:0':ok3_0(+(1, n72703_0)), gen_tt:mark:0':ok3_0(b))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(48)
Obligation:
TRS:
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':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok


Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0)
U13(gen_tt:mark:0':ok3_0(+(1, n3519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n3519_0)
U14(gen_tt:mark:0':ok3_0(+(1, n7631_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n7631_0)
U15(gen_tt:mark:0':ok3_0(+(1, n12352_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n12352_0)
U16(gen_tt:mark:0':ok3_0(+(1, n15653_0))) -> *4_0, rt in Omega(n15653_0)
U22(gen_tt:mark:0':ok3_0(+(1, n17015_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n17015_0)
U23(gen_tt:mark:0':ok3_0(+(1, n20800_0))) -> *4_0, rt in Omega(n20800_0)
U32(gen_tt:mark:0':ok3_0(+(1, n22413_0))) -> *4_0, rt in Omega(n22413_0)
U52(gen_tt:mark:0':ok3_0(+(1, n24127_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n24127_0)
U62(gen_tt:mark:0':ok3_0(+(1, n28634_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n28634_0)
U63(gen_tt:mark:0':ok3_0(+(1, n36295_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n36295_0)
U64(gen_tt:mark:0':ok3_0(+(1, n44565_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n44565_0)
s(gen_tt:mark:0':ok3_0(+(1, n53444_0))) -> *4_0, rt in Omega(n53444_0)
plus(gen_tt:mark:0':ok3_0(+(1, n56006_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n56006_0)
U11(gen_tt:mark:0':ok3_0(+(1, n62438_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n62438_0)
U21(gen_tt:mark:0':ok3_0(+(1, n72703_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n72703_0)


Generator Equations:
gen_tt:mark:0':ok3_0(0) <=> tt
gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(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

----------------------------------------

(49) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U31(gen_tt:mark:0':ok3_0(+(1, n79646_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n79646_0)

Induction Base:
U31(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b))

Induction Step:
U31(gen_tt:mark:0':ok3_0(+(1, +(n79646_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1)
mark(U31(gen_tt:mark:0':ok3_0(+(1, n79646_0)), gen_tt:mark:0':ok3_0(b))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(50)
Obligation:
TRS:
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':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok


Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0)
U13(gen_tt:mark:0':ok3_0(+(1, n3519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n3519_0)
U14(gen_tt:mark:0':ok3_0(+(1, n7631_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n7631_0)
U15(gen_tt:mark:0':ok3_0(+(1, n12352_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n12352_0)
U16(gen_tt:mark:0':ok3_0(+(1, n15653_0))) -> *4_0, rt in Omega(n15653_0)
U22(gen_tt:mark:0':ok3_0(+(1, n17015_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n17015_0)
U23(gen_tt:mark:0':ok3_0(+(1, n20800_0))) -> *4_0, rt in Omega(n20800_0)
U32(gen_tt:mark:0':ok3_0(+(1, n22413_0))) -> *4_0, rt in Omega(n22413_0)
U52(gen_tt:mark:0':ok3_0(+(1, n24127_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n24127_0)
U62(gen_tt:mark:0':ok3_0(+(1, n28634_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n28634_0)
U63(gen_tt:mark:0':ok3_0(+(1, n36295_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n36295_0)
U64(gen_tt:mark:0':ok3_0(+(1, n44565_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n44565_0)
s(gen_tt:mark:0':ok3_0(+(1, n53444_0))) -> *4_0, rt in Omega(n53444_0)
plus(gen_tt:mark:0':ok3_0(+(1, n56006_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n56006_0)
U11(gen_tt:mark:0':ok3_0(+(1, n62438_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n62438_0)
U21(gen_tt:mark:0':ok3_0(+(1, n72703_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n72703_0)
U31(gen_tt:mark:0':ok3_0(+(1, n79646_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n79646_0)


Generator Equations:
gen_tt:mark:0':ok3_0(0) <=> tt
gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(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

----------------------------------------

(51) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U41(gen_tt:mark:0':ok3_0(+(1, n86895_0))) -> *4_0, rt in Omega(n86895_0)

Induction Base:
U41(gen_tt:mark:0':ok3_0(+(1, 0)))

Induction Step:
U41(gen_tt:mark:0':ok3_0(+(1, +(n86895_0, 1)))) ->_R^Omega(1)
mark(U41(gen_tt:mark:0':ok3_0(+(1, n86895_0)))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(52)
Obligation:
TRS:
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':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok


Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0)
U13(gen_tt:mark:0':ok3_0(+(1, n3519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n3519_0)
U14(gen_tt:mark:0':ok3_0(+(1, n7631_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n7631_0)
U15(gen_tt:mark:0':ok3_0(+(1, n12352_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n12352_0)
U16(gen_tt:mark:0':ok3_0(+(1, n15653_0))) -> *4_0, rt in Omega(n15653_0)
U22(gen_tt:mark:0':ok3_0(+(1, n17015_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n17015_0)
U23(gen_tt:mark:0':ok3_0(+(1, n20800_0))) -> *4_0, rt in Omega(n20800_0)
U32(gen_tt:mark:0':ok3_0(+(1, n22413_0))) -> *4_0, rt in Omega(n22413_0)
U52(gen_tt:mark:0':ok3_0(+(1, n24127_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n24127_0)
U62(gen_tt:mark:0':ok3_0(+(1, n28634_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n28634_0)
U63(gen_tt:mark:0':ok3_0(+(1, n36295_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n36295_0)
U64(gen_tt:mark:0':ok3_0(+(1, n44565_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n44565_0)
s(gen_tt:mark:0':ok3_0(+(1, n53444_0))) -> *4_0, rt in Omega(n53444_0)
plus(gen_tt:mark:0':ok3_0(+(1, n56006_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n56006_0)
U11(gen_tt:mark:0':ok3_0(+(1, n62438_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n62438_0)
U21(gen_tt:mark:0':ok3_0(+(1, n72703_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n72703_0)
U31(gen_tt:mark:0':ok3_0(+(1, n79646_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n79646_0)
U41(gen_tt:mark:0':ok3_0(+(1, n86895_0))) -> *4_0, rt in Omega(n86895_0)


Generator Equations:
gen_tt:mark:0':ok3_0(0) <=> tt
gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(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

----------------------------------------

(53) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U51(gen_tt:mark:0':ok3_0(+(1, n90207_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n90207_0)

Induction Base:
U51(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b))

Induction Step:
U51(gen_tt:mark:0':ok3_0(+(1, +(n90207_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1)
mark(U51(gen_tt:mark:0':ok3_0(+(1, n90207_0)), gen_tt:mark:0':ok3_0(b))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(54)
Obligation:
TRS:
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':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok


Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0)
U13(gen_tt:mark:0':ok3_0(+(1, n3519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n3519_0)
U14(gen_tt:mark:0':ok3_0(+(1, n7631_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n7631_0)
U15(gen_tt:mark:0':ok3_0(+(1, n12352_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n12352_0)
U16(gen_tt:mark:0':ok3_0(+(1, n15653_0))) -> *4_0, rt in Omega(n15653_0)
U22(gen_tt:mark:0':ok3_0(+(1, n17015_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n17015_0)
U23(gen_tt:mark:0':ok3_0(+(1, n20800_0))) -> *4_0, rt in Omega(n20800_0)
U32(gen_tt:mark:0':ok3_0(+(1, n22413_0))) -> *4_0, rt in Omega(n22413_0)
U52(gen_tt:mark:0':ok3_0(+(1, n24127_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n24127_0)
U62(gen_tt:mark:0':ok3_0(+(1, n28634_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n28634_0)
U63(gen_tt:mark:0':ok3_0(+(1, n36295_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n36295_0)
U64(gen_tt:mark:0':ok3_0(+(1, n44565_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n44565_0)
s(gen_tt:mark:0':ok3_0(+(1, n53444_0))) -> *4_0, rt in Omega(n53444_0)
plus(gen_tt:mark:0':ok3_0(+(1, n56006_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n56006_0)
U11(gen_tt:mark:0':ok3_0(+(1, n62438_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n62438_0)
U21(gen_tt:mark:0':ok3_0(+(1, n72703_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n72703_0)
U31(gen_tt:mark:0':ok3_0(+(1, n79646_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n79646_0)
U41(gen_tt:mark:0':ok3_0(+(1, n86895_0))) -> *4_0, rt in Omega(n86895_0)
U51(gen_tt:mark:0':ok3_0(+(1, n90207_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n90207_0)


Generator Equations:
gen_tt:mark:0':ok3_0(0) <=> tt
gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(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

----------------------------------------

(55) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U61(gen_tt:mark:0':ok3_0(+(1, n97970_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n97970_0)

Induction Base:
U61(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))

Induction Step:
U61(gen_tt:mark:0':ok3_0(+(1, +(n97970_0, 1))), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) ->_R^Omega(1)
mark(U61(gen_tt:mark:0':ok3_0(+(1, n97970_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(56)
Obligation:
TRS:
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':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok


Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0)
U13(gen_tt:mark:0':ok3_0(+(1, n3519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n3519_0)
U14(gen_tt:mark:0':ok3_0(+(1, n7631_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n7631_0)
U15(gen_tt:mark:0':ok3_0(+(1, n12352_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n12352_0)
U16(gen_tt:mark:0':ok3_0(+(1, n15653_0))) -> *4_0, rt in Omega(n15653_0)
U22(gen_tt:mark:0':ok3_0(+(1, n17015_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n17015_0)
U23(gen_tt:mark:0':ok3_0(+(1, n20800_0))) -> *4_0, rt in Omega(n20800_0)
U32(gen_tt:mark:0':ok3_0(+(1, n22413_0))) -> *4_0, rt in Omega(n22413_0)
U52(gen_tt:mark:0':ok3_0(+(1, n24127_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n24127_0)
U62(gen_tt:mark:0':ok3_0(+(1, n28634_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n28634_0)
U63(gen_tt:mark:0':ok3_0(+(1, n36295_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n36295_0)
U64(gen_tt:mark:0':ok3_0(+(1, n44565_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n44565_0)
s(gen_tt:mark:0':ok3_0(+(1, n53444_0))) -> *4_0, rt in Omega(n53444_0)
plus(gen_tt:mark:0':ok3_0(+(1, n56006_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n56006_0)
U11(gen_tt:mark:0':ok3_0(+(1, n62438_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n62438_0)
U21(gen_tt:mark:0':ok3_0(+(1, n72703_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n72703_0)
U31(gen_tt:mark:0':ok3_0(+(1, n79646_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n79646_0)
U41(gen_tt:mark:0':ok3_0(+(1, n86895_0))) -> *4_0, rt in Omega(n86895_0)
U51(gen_tt:mark:0':ok3_0(+(1, n90207_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n90207_0)
U61(gen_tt:mark:0':ok3_0(+(1, n97970_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n97970_0)


Generator Equations:
gen_tt:mark:0':ok3_0(0) <=> tt
gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(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