KILLED



    


Runtime Complexity (full) proof of /tmp/tmp_Z8YgV/MYNAT_complete_C.xml


(0) Obligation:

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

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0)) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0)) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(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))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
U11(mark(X1), X2, X3) →+ mark(U11(X1, X2, X3))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X1 / mark(X1)].
The result substitution is [ ].

(2) BOUNDS(n^1, INF)

(3) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(4) Obligation:

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

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0')
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0')) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0')) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(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))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0') → ok(0')
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0')
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0')) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0')) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(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))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0') → ok(0')
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
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
isNat :: tt:mark:0':ok → tt:mark:0':ok
U13 :: 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
U31 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U32 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U33 :: tt:mark:0':ok → tt:mark:0':ok
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U51 :: 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
U61 :: tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
U71 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
x :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
isNatKind :: tt:mark:0':ok → 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

(7) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
active, U12, isNat, U13, U22, U32, U33, s, plus, x, U11, and, isNatKind, U21, U31, U41, U51, U61, U71, proper, top

They will be analysed ascendingly in the following order:
U12 < active
isNat < active
U13 < active
U22 < active
U32 < active
U33 < active
s < active
plus < active
x < active
U11 < active
and < active
isNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
active < top
U12 < proper
isNat < proper
U13 < proper
U22 < proper
U32 < proper
U33 < proper
s < proper
plus < proper
x < proper
U11 < proper
and < proper
isNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
proper < top

(8) Obligation:

TRS:
Rules:
active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0')
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0')) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0')) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(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))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0') → ok(0')
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
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
isNat :: tt:mark:0':ok → tt:mark:0':ok
U13 :: 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
U31 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U32 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U33 :: tt:mark:0':ok → tt:mark:0':ok
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U51 :: 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
U61 :: tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
U71 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
x :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
isNatKind :: tt:mark:0':ok → 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, isNat, U13, U22, U32, U33, s, plus, x, U11, and, isNatKind, U21, U31, U41, U51, U61, U71, proper, top

They will be analysed ascendingly in the following order:
U12 < active
isNat < active
U13 < active
U22 < active
U32 < active
U33 < active
s < active
plus < active
x < active
U11 < active
and < active
isNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
active < top
U12 < proper
isNat < proper
U13 < proper
U22 < proper
U32 < proper
U33 < proper
s < proper
plus < proper
x < proper
U11 < proper
and < proper
isNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
proper < top

(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)

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

Induction Step:
U12(gen_tt:mark:0':ok3_0(+(1, +(n5_0, 1))), gen_tt:mark:0':ok3_0(b)) →RΩ(1)
mark(U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b))) →IH
mark(*4_0)

We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).

(10) Complex Obligation (BEST)

(11) Obligation:

TRS:
Rules:
active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0')
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0')) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0')) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(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))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0') → ok(0')
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
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
isNat :: tt:mark:0':ok → tt:mark:0':ok
U13 :: 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
U31 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U32 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U33 :: tt:mark:0':ok → tt:mark:0':ok
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U51 :: 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
U61 :: tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
U71 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
x :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
isNatKind :: tt:mark:0':ok → 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)) → *4_0, rt ∈ Ω(n50)

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, U13, U22, U32, U33, s, plus, x, U11, and, isNatKind, U21, U31, U41, U51, U61, U71, proper, top

They will be analysed ascendingly in the following order:
isNat < active
U13 < active
U22 < active
U32 < active
U33 < active
s < active
plus < active
x < active
U11 < active
and < active
isNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
active < top
isNat < proper
U13 < proper
U22 < proper
U32 < proper
U33 < proper
s < proper
plus < proper
x < proper
U11 < proper
and < proper
isNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
proper < top

(12) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(13) Obligation:

TRS:
Rules:
active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0')
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0')) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0')) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(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))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0') → ok(0')
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
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
isNat :: tt:mark:0':ok → tt:mark:0':ok
U13 :: 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
U31 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U32 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U33 :: tt:mark:0':ok → tt:mark:0':ok
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U51 :: 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
U61 :: tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
U71 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
x :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
isNatKind :: tt:mark:0':ok → 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)) → *4_0, rt ∈ Ω(n50)

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:
U13, active, U22, U32, U33, s, plus, x, U11, and, isNatKind, U21, U31, U41, U51, U61, U71, proper, top

They will be analysed ascendingly in the following order:
U13 < active
U22 < active
U32 < active
U33 < active
s < active
plus < active
x < active
U11 < active
and < active
isNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
active < top
U13 < proper
U22 < proper
U32 < proper
U33 < proper
s < proper
plus < proper
x < proper
U11 < proper
and < proper
isNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
proper < top

(14) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
U13(gen_tt:mark:0':ok3_0(+(1, n1763_0))) → *4_0, rt ∈ Ω(n17630)

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

Induction Step:
U13(gen_tt:mark:0':ok3_0(+(1, +(n1763_0, 1)))) →RΩ(1)
mark(U13(gen_tt:mark:0':ok3_0(+(1, n1763_0)))) →IH
mark(*4_0)

We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).

(15) Complex Obligation (BEST)

(16) Obligation:

TRS:
Rules:
active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0')
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0')) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0')) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(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))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0') → ok(0')
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
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
isNat :: tt:mark:0':ok → tt:mark:0':ok
U13 :: 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
U31 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U32 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U33 :: tt:mark:0':ok → tt:mark:0':ok
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U51 :: 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
U61 :: tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
U71 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
x :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
isNatKind :: tt:mark:0':ok → 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)) → *4_0, rt ∈ Ω(n50)
U13(gen_tt:mark:0':ok3_0(+(1, n1763_0))) → *4_0, rt ∈ Ω(n17630)

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, U32, U33, s, plus, x, U11, and, isNatKind, U21, U31, U41, U51, U61, U71, proper, top

They will be analysed ascendingly in the following order:
U22 < active
U32 < active
U33 < active
s < active
plus < active
x < active
U11 < active
and < active
isNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
active < top
U22 < proper
U32 < proper
U33 < proper
s < proper
plus < proper
x < proper
U11 < proper
and < proper
isNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
proper < top

(17) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
U22(gen_tt:mark:0':ok3_0(+(1, n2483_0))) → *4_0, rt ∈ Ω(n24830)

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

Induction Step:
U22(gen_tt:mark:0':ok3_0(+(1, +(n2483_0, 1)))) →RΩ(1)
mark(U22(gen_tt:mark:0':ok3_0(+(1, n2483_0)))) →IH
mark(*4_0)

We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).

(18) Complex Obligation (BEST)

(19) Obligation:

TRS:
Rules:
active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0')
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0')) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0')) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(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))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0') → ok(0')
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
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
isNat :: tt:mark:0':ok → tt:mark:0':ok
U13 :: 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
U31 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U32 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U33 :: tt:mark:0':ok → tt:mark:0':ok
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U51 :: 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
U61 :: tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
U71 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
x :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
isNatKind :: tt:mark:0':ok → 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)) → *4_0, rt ∈ Ω(n50)
U13(gen_tt:mark:0':ok3_0(+(1, n1763_0))) → *4_0, rt ∈ Ω(n17630)
U22(gen_tt:mark:0':ok3_0(+(1, n2483_0))) → *4_0, rt ∈ Ω(n24830)

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, U33, s, plus, x, U11, and, isNatKind, U21, U31, U41, U51, U61, U71, proper, top

They will be analysed ascendingly in the following order:
U32 < active
U33 < active
s < active
plus < active
x < active
U11 < active
and < active
isNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
active < top
U32 < proper
U33 < proper
s < proper
plus < proper
x < proper
U11 < proper
and < proper
isNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
proper < top

(20) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
U32(gen_tt:mark:0':ok3_0(+(1, n3304_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n33040)

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

Induction Step:
U32(gen_tt:mark:0':ok3_0(+(1, +(n3304_0, 1))), gen_tt:mark:0':ok3_0(b)) →RΩ(1)
mark(U32(gen_tt:mark:0':ok3_0(+(1, n3304_0)), gen_tt:mark:0':ok3_0(b))) →IH
mark(*4_0)

We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).

(21) Complex Obligation (BEST)

(22) Obligation:

TRS:
Rules:
active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0')
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0')) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0')) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(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))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0') → ok(0')
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
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
isNat :: tt:mark:0':ok → tt:mark:0':ok
U13 :: 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
U31 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U32 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U33 :: tt:mark:0':ok → tt:mark:0':ok
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U51 :: 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
U61 :: tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
U71 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
x :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
isNatKind :: tt:mark:0':ok → 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)) → *4_0, rt ∈ Ω(n50)
U13(gen_tt:mark:0':ok3_0(+(1, n1763_0))) → *4_0, rt ∈ Ω(n17630)
U22(gen_tt:mark:0':ok3_0(+(1, n2483_0))) → *4_0, rt ∈ Ω(n24830)
U32(gen_tt:mark:0':ok3_0(+(1, n3304_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n33040)

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:
U33, active, s, plus, x, U11, and, isNatKind, U21, U31, U41, U51, U61, U71, proper, top

They will be analysed ascendingly in the following order:
U33 < active
s < active
plus < active
x < active
U11 < active
and < active
isNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
active < top
U33 < proper
s < proper
plus < proper
x < proper
U11 < proper
and < proper
isNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
proper < top

(23) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
U33(gen_tt:mark:0':ok3_0(+(1, n5769_0))) → *4_0, rt ∈ Ω(n57690)

Induction Base:
U33(gen_tt:mark:0':ok3_0(+(1, 0)))

Induction Step:
U33(gen_tt:mark:0':ok3_0(+(1, +(n5769_0, 1)))) →RΩ(1)
mark(U33(gen_tt:mark:0':ok3_0(+(1, n5769_0)))) →IH
mark(*4_0)

We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).

(24) Complex Obligation (BEST)

(25) Obligation:

TRS:
Rules:
active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0')
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0')) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0')) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(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))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0') → ok(0')
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
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
isNat :: tt:mark:0':ok → tt:mark:0':ok
U13 :: 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
U31 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U32 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U33 :: tt:mark:0':ok → tt:mark:0':ok
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U51 :: 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
U61 :: tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
U71 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
x :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
isNatKind :: tt:mark:0':ok → 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)) → *4_0, rt ∈ Ω(n50)
U13(gen_tt:mark:0':ok3_0(+(1, n1763_0))) → *4_0, rt ∈ Ω(n17630)
U22(gen_tt:mark:0':ok3_0(+(1, n2483_0))) → *4_0, rt ∈ Ω(n24830)
U32(gen_tt:mark:0':ok3_0(+(1, n3304_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n33040)
U33(gen_tt:mark:0':ok3_0(+(1, n5769_0))) → *4_0, rt ∈ Ω(n57690)

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, x, U11, and, isNatKind, U21, U31, U41, U51, U61, U71, proper, top

They will be analysed ascendingly in the following order:
s < active
plus < active
x < active
U11 < active
and < active
isNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
active < top
s < proper
plus < proper
x < proper
U11 < proper
and < proper
isNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
proper < top

(26) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
s(gen_tt:mark:0':ok3_0(+(1, n6841_0))) → *4_0, rt ∈ Ω(n68410)

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

Induction Step:
s(gen_tt:mark:0':ok3_0(+(1, +(n6841_0, 1)))) →RΩ(1)
mark(s(gen_tt:mark:0':ok3_0(+(1, n6841_0)))) →IH
mark(*4_0)

We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).

(27) Complex Obligation (BEST)

(28) Obligation:

TRS:
Rules:
active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0')
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0')) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0')) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(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))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0') → ok(0')
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
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
isNat :: tt:mark:0':ok → tt:mark:0':ok
U13 :: 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
U31 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U32 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U33 :: tt:mark:0':ok → tt:mark:0':ok
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U51 :: 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
U61 :: tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
U71 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
x :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
isNatKind :: tt:mark:0':ok → 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)) → *4_0, rt ∈ Ω(n50)
U13(gen_tt:mark:0':ok3_0(+(1, n1763_0))) → *4_0, rt ∈ Ω(n17630)
U22(gen_tt:mark:0':ok3_0(+(1, n2483_0))) → *4_0, rt ∈ Ω(n24830)
U32(gen_tt:mark:0':ok3_0(+(1, n3304_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n33040)
U33(gen_tt:mark:0':ok3_0(+(1, n5769_0))) → *4_0, rt ∈ Ω(n57690)
s(gen_tt:mark:0':ok3_0(+(1, n6841_0))) → *4_0, rt ∈ Ω(n68410)

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, x, U11, and, isNatKind, U21, U31, U41, U51, U61, U71, proper, top

They will be analysed ascendingly in the following order:
plus < active
x < active
U11 < active
and < active
isNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
active < top
plus < proper
x < proper
U11 < proper
and < proper
isNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
proper < top

(29) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
plus(gen_tt:mark:0':ok3_0(+(1, n8014_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n80140)

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, +(n8014_0, 1))), gen_tt:mark:0':ok3_0(b)) →RΩ(1)
mark(plus(gen_tt:mark:0':ok3_0(+(1, n8014_0)), gen_tt:mark:0':ok3_0(b))) →IH
mark(*4_0)

We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).

(30) Complex Obligation (BEST)

(31) Obligation:

TRS:
Rules:
active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0')
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0')) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0')) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(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))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0') → ok(0')
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
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
isNat :: tt:mark:0':ok → tt:mark:0':ok
U13 :: 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
U31 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U32 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U33 :: tt:mark:0':ok → tt:mark:0':ok
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U51 :: 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
U61 :: tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
U71 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
x :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
isNatKind :: tt:mark:0':ok → 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)) → *4_0, rt ∈ Ω(n50)
U13(gen_tt:mark:0':ok3_0(+(1, n1763_0))) → *4_0, rt ∈ Ω(n17630)
U22(gen_tt:mark:0':ok3_0(+(1, n2483_0))) → *4_0, rt ∈ Ω(n24830)
U32(gen_tt:mark:0':ok3_0(+(1, n3304_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n33040)
U33(gen_tt:mark:0':ok3_0(+(1, n5769_0))) → *4_0, rt ∈ Ω(n57690)
s(gen_tt:mark:0':ok3_0(+(1, n6841_0))) → *4_0, rt ∈ Ω(n68410)
plus(gen_tt:mark:0':ok3_0(+(1, n8014_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n80140)

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:
x, active, U11, and, isNatKind, U21, U31, U41, U51, U61, U71, proper, top

They will be analysed ascendingly in the following order:
x < active
U11 < active
and < active
isNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
active < top
x < proper
U11 < proper
and < proper
isNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
proper < top

(32) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
x(gen_tt:mark:0':ok3_0(+(1, n11394_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n113940)

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

Induction Step:
x(gen_tt:mark:0':ok3_0(+(1, +(n11394_0, 1))), gen_tt:mark:0':ok3_0(b)) →RΩ(1)
mark(x(gen_tt:mark:0':ok3_0(+(1, n11394_0)), gen_tt:mark:0':ok3_0(b))) →IH
mark(*4_0)

We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).

(33) Complex Obligation (BEST)

(34) Obligation:

TRS:
Rules:
active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0')
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0')) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0')) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(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))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0') → ok(0')
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
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
isNat :: tt:mark:0':ok → tt:mark:0':ok
U13 :: 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
U31 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U32 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U33 :: tt:mark:0':ok → tt:mark:0':ok
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U51 :: 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
U61 :: tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
U71 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
x :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
isNatKind :: tt:mark:0':ok → 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)) → *4_0, rt ∈ Ω(n50)
U13(gen_tt:mark:0':ok3_0(+(1, n1763_0))) → *4_0, rt ∈ Ω(n17630)
U22(gen_tt:mark:0':ok3_0(+(1, n2483_0))) → *4_0, rt ∈ Ω(n24830)
U32(gen_tt:mark:0':ok3_0(+(1, n3304_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n33040)
U33(gen_tt:mark:0':ok3_0(+(1, n5769_0))) → *4_0, rt ∈ Ω(n57690)
s(gen_tt:mark:0':ok3_0(+(1, n6841_0))) → *4_0, rt ∈ Ω(n68410)
plus(gen_tt:mark:0':ok3_0(+(1, n8014_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n80140)
x(gen_tt:mark:0':ok3_0(+(1, n11394_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n113940)

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, and, isNatKind, U21, U31, U41, U51, U61, U71, proper, top

They will be analysed ascendingly in the following order:
U11 < active
and < active
isNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
active < top
U11 < proper
and < proper
isNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
proper < top

(35) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
U11(gen_tt:mark:0':ok3_0(+(1, n15078_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n150780)

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, +(n15078_0, 1))), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) →RΩ(1)
mark(U11(gen_tt:mark:0':ok3_0(+(1, n15078_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))) →IH
mark(*4_0)

We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).

(36) Complex Obligation (BEST)

(37) Obligation:

TRS:
Rules:
active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0')
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0')) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0')) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(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))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0') → ok(0')
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
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
isNat :: tt:mark:0':ok → tt:mark:0':ok
U13 :: 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
U31 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U32 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U33 :: tt:mark:0':ok → tt:mark:0':ok
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U51 :: 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
U61 :: tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
U71 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
x :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
isNatKind :: tt:mark:0':ok → 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)) → *4_0, rt ∈ Ω(n50)
U13(gen_tt:mark:0':ok3_0(+(1, n1763_0))) → *4_0, rt ∈ Ω(n17630)
U22(gen_tt:mark:0':ok3_0(+(1, n2483_0))) → *4_0, rt ∈ Ω(n24830)
U32(gen_tt:mark:0':ok3_0(+(1, n3304_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n33040)
U33(gen_tt:mark:0':ok3_0(+(1, n5769_0))) → *4_0, rt ∈ Ω(n57690)
s(gen_tt:mark:0':ok3_0(+(1, n6841_0))) → *4_0, rt ∈ Ω(n68410)
plus(gen_tt:mark:0':ok3_0(+(1, n8014_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n80140)
x(gen_tt:mark:0':ok3_0(+(1, n11394_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n113940)
U11(gen_tt:mark:0':ok3_0(+(1, n15078_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n150780)

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:
and, active, isNatKind, U21, U31, U41, U51, U61, U71, proper, top

They will be analysed ascendingly in the following order:
and < active
isNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
active < top
and < proper
isNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
proper < top

(38) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
and(gen_tt:mark:0':ok3_0(+(1, n21186_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n211860)

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

Induction Step:
and(gen_tt:mark:0':ok3_0(+(1, +(n21186_0, 1))), gen_tt:mark:0':ok3_0(b)) →RΩ(1)
mark(and(gen_tt:mark:0':ok3_0(+(1, n21186_0)), gen_tt:mark:0':ok3_0(b))) →IH
mark(*4_0)

We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).

(39) Complex Obligation (BEST)

(40) Obligation:

TRS:
Rules:
active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0')
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0')) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0')) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(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))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0') → ok(0')
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
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
isNat :: tt:mark:0':ok → tt:mark:0':ok
U13 :: 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
U31 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U32 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U33 :: tt:mark:0':ok → tt:mark:0':ok
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U51 :: 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
U61 :: tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
U71 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
x :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
isNatKind :: tt:mark:0':ok → 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)) → *4_0, rt ∈ Ω(n50)
U13(gen_tt:mark:0':ok3_0(+(1, n1763_0))) → *4_0, rt ∈ Ω(n17630)
U22(gen_tt:mark:0':ok3_0(+(1, n2483_0))) → *4_0, rt ∈ Ω(n24830)
U32(gen_tt:mark:0':ok3_0(+(1, n3304_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n33040)
U33(gen_tt:mark:0':ok3_0(+(1, n5769_0))) → *4_0, rt ∈ Ω(n57690)
s(gen_tt:mark:0':ok3_0(+(1, n6841_0))) → *4_0, rt ∈ Ω(n68410)
plus(gen_tt:mark:0':ok3_0(+(1, n8014_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n80140)
x(gen_tt:mark:0':ok3_0(+(1, n11394_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n113940)
U11(gen_tt:mark:0':ok3_0(+(1, n15078_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n150780)
and(gen_tt:mark:0':ok3_0(+(1, n21186_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n211860)

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, U21, U31, U41, U51, U61, U71, proper, top

They will be analysed ascendingly in the following order:
isNatKind < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
active < top
isNatKind < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
proper < top

(41) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(42) Obligation:

TRS:
Rules:
active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0')
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0')) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0')) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(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))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0') → ok(0')
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
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
isNat :: tt:mark:0':ok → tt:mark:0':ok
U13 :: 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
U31 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U32 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U33 :: tt:mark:0':ok → tt:mark:0':ok
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U51 :: 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
U61 :: tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
U71 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
x :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
isNatKind :: tt:mark:0':ok → 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)) → *4_0, rt ∈ Ω(n50)
U13(gen_tt:mark:0':ok3_0(+(1, n1763_0))) → *4_0, rt ∈ Ω(n17630)
U22(gen_tt:mark:0':ok3_0(+(1, n2483_0))) → *4_0, rt ∈ Ω(n24830)
U32(gen_tt:mark:0':ok3_0(+(1, n3304_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n33040)
U33(gen_tt:mark:0':ok3_0(+(1, n5769_0))) → *4_0, rt ∈ Ω(n57690)
s(gen_tt:mark:0':ok3_0(+(1, n6841_0))) → *4_0, rt ∈ Ω(n68410)
plus(gen_tt:mark:0':ok3_0(+(1, n8014_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n80140)
x(gen_tt:mark:0':ok3_0(+(1, n11394_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n113940)
U11(gen_tt:mark:0':ok3_0(+(1, n15078_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n150780)
and(gen_tt:mark:0':ok3_0(+(1, n21186_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n211860)

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, U71, proper, top

They will be analysed ascendingly in the following order:
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
active < top
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
proper < top

(43) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
U21(gen_tt:mark:0':ok3_0(+(1, n25425_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n254250)

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, +(n25425_0, 1))), gen_tt:mark:0':ok3_0(b)) →RΩ(1)
mark(U21(gen_tt:mark:0':ok3_0(+(1, n25425_0)), gen_tt:mark:0':ok3_0(b))) →IH
mark(*4_0)

We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).

(44) Complex Obligation (BEST)

(45) Obligation:

TRS:
Rules:
active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0')
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0')) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0')) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(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))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0') → ok(0')
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
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
isNat :: tt:mark:0':ok → tt:mark:0':ok
U13 :: 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
U31 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U32 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U33 :: tt:mark:0':ok → tt:mark:0':ok
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U51 :: 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
U61 :: tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
U71 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
x :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
isNatKind :: tt:mark:0':ok → 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)) → *4_0, rt ∈ Ω(n50)
U13(gen_tt:mark:0':ok3_0(+(1, n1763_0))) → *4_0, rt ∈ Ω(n17630)
U22(gen_tt:mark:0':ok3_0(+(1, n2483_0))) → *4_0, rt ∈ Ω(n24830)
U32(gen_tt:mark:0':ok3_0(+(1, n3304_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n33040)
U33(gen_tt:mark:0':ok3_0(+(1, n5769_0))) → *4_0, rt ∈ Ω(n57690)
s(gen_tt:mark:0':ok3_0(+(1, n6841_0))) → *4_0, rt ∈ Ω(n68410)
plus(gen_tt:mark:0':ok3_0(+(1, n8014_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n80140)
x(gen_tt:mark:0':ok3_0(+(1, n11394_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n113940)
U11(gen_tt:mark:0':ok3_0(+(1, n15078_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n150780)
and(gen_tt:mark:0':ok3_0(+(1, n21186_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n211860)
U21(gen_tt:mark:0':ok3_0(+(1, n25425_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n254250)

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, U71, proper, top

They will be analysed ascendingly in the following order:
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
active < top
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
proper < top

(46) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
U31(gen_tt:mark:0':ok3_0(+(1, n29920_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299200)

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

Induction Step:
U31(gen_tt:mark:0':ok3_0(+(1, +(n29920_0, 1))), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) →RΩ(1)
mark(U31(gen_tt:mark:0':ok3_0(+(1, n29920_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))) →IH
mark(*4_0)

We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).

(47) Complex Obligation (BEST)

(48) Obligation:

TRS:
Rules:
active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0')
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0')) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0')) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(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))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0') → ok(0')
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
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
isNat :: tt:mark:0':ok → tt:mark:0':ok
U13 :: 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
U31 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U32 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U33 :: tt:mark:0':ok → tt:mark:0':ok
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U51 :: 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
U61 :: tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
U71 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
x :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
isNatKind :: tt:mark:0':ok → 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)) → *4_0, rt ∈ Ω(n50)
U13(gen_tt:mark:0':ok3_0(+(1, n1763_0))) → *4_0, rt ∈ Ω(n17630)
U22(gen_tt:mark:0':ok3_0(+(1, n2483_0))) → *4_0, rt ∈ Ω(n24830)
U32(gen_tt:mark:0':ok3_0(+(1, n3304_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n33040)
U33(gen_tt:mark:0':ok3_0(+(1, n5769_0))) → *4_0, rt ∈ Ω(n57690)
s(gen_tt:mark:0':ok3_0(+(1, n6841_0))) → *4_0, rt ∈ Ω(n68410)
plus(gen_tt:mark:0':ok3_0(+(1, n8014_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n80140)
x(gen_tt:mark:0':ok3_0(+(1, n11394_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n113940)
U11(gen_tt:mark:0':ok3_0(+(1, n15078_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n150780)
and(gen_tt:mark:0':ok3_0(+(1, n21186_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n211860)
U21(gen_tt:mark:0':ok3_0(+(1, n25425_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n254250)
U31(gen_tt:mark:0':ok3_0(+(1, n29920_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299200)

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, U71, proper, top

They will be analysed ascendingly in the following order:
U41 < active
U51 < active
U61 < active
U71 < active
active < top
U41 < proper
U51 < proper
U61 < proper
U71 < proper
proper < top

(49) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
U41(gen_tt:mark:0':ok3_0(+(1, n37549_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n375490)

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

Induction Step:
U41(gen_tt:mark:0':ok3_0(+(1, +(n37549_0, 1))), gen_tt:mark:0':ok3_0(b)) →RΩ(1)
mark(U41(gen_tt:mark:0':ok3_0(+(1, n37549_0)), gen_tt:mark:0':ok3_0(b))) →IH
mark(*4_0)

We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).

(50) Complex Obligation (BEST)

(51) Obligation:

TRS:
Rules:
active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0')
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0')) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0')) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(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))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0') → ok(0')
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
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
isNat :: tt:mark:0':ok → tt:mark:0':ok
U13 :: 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
U31 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U32 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U33 :: tt:mark:0':ok → tt:mark:0':ok
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U51 :: 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
U61 :: tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
U71 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
x :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
isNatKind :: tt:mark:0':ok → 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)) → *4_0, rt ∈ Ω(n50)
U13(gen_tt:mark:0':ok3_0(+(1, n1763_0))) → *4_0, rt ∈ Ω(n17630)
U22(gen_tt:mark:0':ok3_0(+(1, n2483_0))) → *4_0, rt ∈ Ω(n24830)
U32(gen_tt:mark:0':ok3_0(+(1, n3304_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n33040)
U33(gen_tt:mark:0':ok3_0(+(1, n5769_0))) → *4_0, rt ∈ Ω(n57690)
s(gen_tt:mark:0':ok3_0(+(1, n6841_0))) → *4_0, rt ∈ Ω(n68410)
plus(gen_tt:mark:0':ok3_0(+(1, n8014_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n80140)
x(gen_tt:mark:0':ok3_0(+(1, n11394_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n113940)
U11(gen_tt:mark:0':ok3_0(+(1, n15078_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n150780)
and(gen_tt:mark:0':ok3_0(+(1, n21186_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n211860)
U21(gen_tt:mark:0':ok3_0(+(1, n25425_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n254250)
U31(gen_tt:mark:0':ok3_0(+(1, n29920_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299200)
U41(gen_tt:mark:0':ok3_0(+(1, n37549_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n375490)

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, U71, proper, top

They will be analysed ascendingly in the following order:
U51 < active
U61 < active
U71 < active
active < top
U51 < proper
U61 < proper
U71 < proper
proper < top

(52) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
U51(gen_tt:mark:0':ok3_0(+(1, n42750_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n427500)

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

Induction Step:
U51(gen_tt:mark:0':ok3_0(+(1, +(n42750_0, 1))), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) →RΩ(1)
mark(U51(gen_tt:mark:0':ok3_0(+(1, n42750_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))) →IH
mark(*4_0)

We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).

(53) Complex Obligation (BEST)

(54) Obligation:

TRS:
Rules:
active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0')
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0')) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0')) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(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))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0') → ok(0')
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
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
isNat :: tt:mark:0':ok → tt:mark:0':ok
U13 :: 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
U31 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U32 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U33 :: tt:mark:0':ok → tt:mark:0':ok
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U51 :: 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
U61 :: tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
U71 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
x :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
isNatKind :: tt:mark:0':ok → 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)) → *4_0, rt ∈ Ω(n50)
U13(gen_tt:mark:0':ok3_0(+(1, n1763_0))) → *4_0, rt ∈ Ω(n17630)
U22(gen_tt:mark:0':ok3_0(+(1, n2483_0))) → *4_0, rt ∈ Ω(n24830)
U32(gen_tt:mark:0':ok3_0(+(1, n3304_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n33040)
U33(gen_tt:mark:0':ok3_0(+(1, n5769_0))) → *4_0, rt ∈ Ω(n57690)
s(gen_tt:mark:0':ok3_0(+(1, n6841_0))) → *4_0, rt ∈ Ω(n68410)
plus(gen_tt:mark:0':ok3_0(+(1, n8014_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n80140)
x(gen_tt:mark:0':ok3_0(+(1, n11394_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n113940)
U11(gen_tt:mark:0':ok3_0(+(1, n15078_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n150780)
and(gen_tt:mark:0':ok3_0(+(1, n21186_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n211860)
U21(gen_tt:mark:0':ok3_0(+(1, n25425_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n254250)
U31(gen_tt:mark:0':ok3_0(+(1, n29920_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299200)
U41(gen_tt:mark:0':ok3_0(+(1, n37549_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n375490)
U51(gen_tt:mark:0':ok3_0(+(1, n42750_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n427500)

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, U71, proper, top

They will be analysed ascendingly in the following order:
U61 < active
U71 < active
active < top
U61 < proper
U71 < proper
proper < top

(55) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
U61(gen_tt:mark:0':ok3_0(+(1, n51442_0))) → *4_0, rt ∈ Ω(n514420)

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

Induction Step:
U61(gen_tt:mark:0':ok3_0(+(1, +(n51442_0, 1)))) →RΩ(1)
mark(U61(gen_tt:mark:0':ok3_0(+(1, n51442_0)))) →IH
mark(*4_0)

We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).

(56) Complex Obligation (BEST)

(57) Obligation:

TRS:
Rules:
active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0')
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0')) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0')) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(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))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0') → ok(0')
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
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
isNat :: tt:mark:0':ok → tt:mark:0':ok
U13 :: 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
U31 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U32 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U33 :: tt:mark:0':ok → tt:mark:0':ok
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U51 :: 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
U61 :: tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
U71 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
x :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
isNatKind :: tt:mark:0':ok → 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)) → *4_0, rt ∈ Ω(n50)
U13(gen_tt:mark:0':ok3_0(+(1, n1763_0))) → *4_0, rt ∈ Ω(n17630)
U22(gen_tt:mark:0':ok3_0(+(1, n2483_0))) → *4_0, rt ∈ Ω(n24830)
U32(gen_tt:mark:0':ok3_0(+(1, n3304_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n33040)
U33(gen_tt:mark:0':ok3_0(+(1, n5769_0))) → *4_0, rt ∈ Ω(n57690)
s(gen_tt:mark:0':ok3_0(+(1, n6841_0))) → *4_0, rt ∈ Ω(n68410)
plus(gen_tt:mark:0':ok3_0(+(1, n8014_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n80140)
x(gen_tt:mark:0':ok3_0(+(1, n11394_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n113940)
U11(gen_tt:mark:0':ok3_0(+(1, n15078_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n150780)
and(gen_tt:mark:0':ok3_0(+(1, n21186_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n211860)
U21(gen_tt:mark:0':ok3_0(+(1, n25425_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n254250)
U31(gen_tt:mark:0':ok3_0(+(1, n29920_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299200)
U41(gen_tt:mark:0':ok3_0(+(1, n37549_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n375490)
U51(gen_tt:mark:0':ok3_0(+(1, n42750_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n427500)
U61(gen_tt:mark:0':ok3_0(+(1, n51442_0))) → *4_0, rt ∈ Ω(n514420)

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:
U71, active, proper, top

They will be analysed ascendingly in the following order:
U71 < active
active < top
U71 < proper
proper < top

(58) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
U71(gen_tt:mark:0':ok3_0(+(1, n54063_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n540630)

Induction Base:
U71(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))

Induction Step:
U71(gen_tt:mark:0':ok3_0(+(1, +(n54063_0, 1))), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) →RΩ(1)
mark(U71(gen_tt:mark:0':ok3_0(+(1, n54063_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))) →IH
mark(*4_0)

We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).

(59) Complex Obligation (BEST)

(60) Obligation:

TRS:
Rules:
active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0')
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0')) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0')) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(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))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0') → ok(0')
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
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
isNat :: tt:mark:0':ok → tt:mark:0':ok
U13 :: 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
U31 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U32 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U33 :: tt:mark:0':ok → tt:mark:0':ok
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U51 :: 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
U61 :: tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
U71 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
x :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
isNatKind :: tt:mark:0':ok → 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)) → *4_0, rt ∈ Ω(n50)
U13(gen_tt:mark:0':ok3_0(+(1, n1763_0))) → *4_0, rt ∈ Ω(n17630)
U22(gen_tt:mark:0':ok3_0(+(1, n2483_0))) → *4_0, rt ∈ Ω(n24830)
U32(gen_tt:mark:0':ok3_0(+(1, n3304_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n33040)
U33(gen_tt:mark:0':ok3_0(+(1, n5769_0))) → *4_0, rt ∈ Ω(n57690)
s(gen_tt:mark:0':ok3_0(+(1, n6841_0))) → *4_0, rt ∈ Ω(n68410)
plus(gen_tt:mark:0':ok3_0(+(1, n8014_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n80140)
x(gen_tt:mark:0':ok3_0(+(1, n11394_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n113940)
U11(gen_tt:mark:0':ok3_0(+(1, n15078_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n150780)
and(gen_tt:mark:0':ok3_0(+(1, n21186_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n211860)
U21(gen_tt:mark:0':ok3_0(+(1, n25425_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n254250)
U31(gen_tt:mark:0':ok3_0(+(1, n29920_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299200)
U41(gen_tt:mark:0':ok3_0(+(1, n37549_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n375490)
U51(gen_tt:mark:0':ok3_0(+(1, n42750_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n427500)
U61(gen_tt:mark:0':ok3_0(+(1, n51442_0))) → *4_0, rt ∈ Ω(n514420)
U71(gen_tt:mark:0':ok3_0(+(1, n54063_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n540630)

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

(61) Obligation:

TRS:
Rules:
active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0')
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0')) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0')) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(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))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0') → ok(0')
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
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
isNat :: tt:mark:0':ok → tt:mark:0':ok
U13 :: 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
U31 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U32 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U33 :: tt:mark:0':ok → tt:mark:0':ok
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U51 :: 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
U61 :: tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
U71 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
x :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
isNatKind :: tt:mark:0':ok → 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)) → *4_0, rt ∈ Ω(n50)
U13(gen_tt:mark:0':ok3_0(+(1, n1763_0))) → *4_0, rt ∈ Ω(n17630)
U22(gen_tt:mark:0':ok3_0(+(1, n2483_0))) → *4_0, rt ∈ Ω(n24830)
U32(gen_tt:mark:0':ok3_0(+(1, n3304_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n33040)
U33(gen_tt:mark:0':ok3_0(+(1, n5769_0))) → *4_0, rt ∈ Ω(n57690)
s(gen_tt:mark:0':ok3_0(+(1, n6841_0))) → *4_0, rt ∈ Ω(n68410)
plus(gen_tt:mark:0':ok3_0(+(1, n8014_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n80140)
x(gen_tt:mark:0':ok3_0(+(1, n11394_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n113940)
U11(gen_tt:mark:0':ok3_0(+(1, n15078_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n150780)
and(gen_tt:mark:0':ok3_0(+(1, n21186_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n211860)
U21(gen_tt:mark:0':ok3_0(+(1, n25425_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n254250)
U31(gen_tt:mark:0':ok3_0(+(1, n29920_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299200)
U41(gen_tt:mark:0':ok3_0(+(1, n37549_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n375490)
U51(gen_tt:mark:0':ok3_0(+(1, n42750_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n427500)
U61(gen_tt:mark:0':ok3_0(+(1, n51442_0))) → *4_0, rt ∈ Ω(n514420)
U71(gen_tt:mark:0':ok3_0(+(1, n54063_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n540630)

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))

No more defined symbols left to analyse.

(62) Obligation:

TRS:
Rules:
active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0')
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0')) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0')) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(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))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0') → ok(0')
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
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
isNat :: tt:mark:0':ok → tt:mark:0':ok
U13 :: 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
U31 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U32 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U33 :: tt:mark:0':ok → tt:mark:0':ok
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U51 :: 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
U61 :: tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
U71 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
x :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
isNatKind :: tt:mark:0':ok → 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)) → *4_0, rt ∈ Ω(n50)
U13(gen_tt:mark:0':ok3_0(+(1, n1763_0))) → *4_0, rt ∈ Ω(n17630)
U22(gen_tt:mark:0':ok3_0(+(1, n2483_0))) → *4_0, rt ∈ Ω(n24830)
U32(gen_tt:mark:0':ok3_0(+(1, n3304_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n33040)
U33(gen_tt:mark:0':ok3_0(+(1, n5769_0))) → *4_0, rt ∈ Ω(n57690)
s(gen_tt:mark:0':ok3_0(+(1, n6841_0))) → *4_0, rt ∈ Ω(n68410)
plus(gen_tt:mark:0':ok3_0(+(1, n8014_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n80140)
x(gen_tt:mark:0':ok3_0(+(1, n11394_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n113940)
U11(gen_tt:mark:0':ok3_0(+(1, n15078_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n150780)
and(gen_tt:mark:0':ok3_0(+(1, n21186_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n211860)
U21(gen_tt:mark:0':ok3_0(+(1, n25425_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n254250)
U31(gen_tt:mark:0':ok3_0(+(1, n29920_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299200)
U41(gen_tt:mark:0':ok3_0(+(1, n37549_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n375490)
U51(gen_tt:mark:0':ok3_0(+(1, n42750_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n427500)
U61(gen_tt:mark:0':ok3_0(+(1, n51442_0))) → *4_0, rt ∈ Ω(n514420)

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))

No more defined symbols left to analyse.

(63) Obligation:

TRS:
Rules:
active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0')
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0')) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0')) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(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))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0') → ok(0')
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
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
isNat :: tt:mark:0':ok → tt:mark:0':ok
U13 :: 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
U31 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U32 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U33 :: tt:mark:0':ok → tt:mark:0':ok
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U51 :: 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
U61 :: tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
U71 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
x :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
isNatKind :: tt:mark:0':ok → 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)) → *4_0, rt ∈ Ω(n50)
U13(gen_tt:mark:0':ok3_0(+(1, n1763_0))) → *4_0, rt ∈ Ω(n17630)
U22(gen_tt:mark:0':ok3_0(+(1, n2483_0))) → *4_0, rt ∈ Ω(n24830)
U32(gen_tt:mark:0':ok3_0(+(1, n3304_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n33040)
U33(gen_tt:mark:0':ok3_0(+(1, n5769_0))) → *4_0, rt ∈ Ω(n57690)
s(gen_tt:mark:0':ok3_0(+(1, n6841_0))) → *4_0, rt ∈ Ω(n68410)
plus(gen_tt:mark:0':ok3_0(+(1, n8014_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n80140)
x(gen_tt:mark:0':ok3_0(+(1, n11394_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n113940)
U11(gen_tt:mark:0':ok3_0(+(1, n15078_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n150780)
and(gen_tt:mark:0':ok3_0(+(1, n21186_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n211860)
U21(gen_tt:mark:0':ok3_0(+(1, n25425_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n254250)
U31(gen_tt:mark:0':ok3_0(+(1, n29920_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299200)
U41(gen_tt:mark:0':ok3_0(+(1, n37549_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n375490)
U51(gen_tt:mark:0':ok3_0(+(1, n42750_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n427500)

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))

No more defined symbols left to analyse.

(64) Obligation:

TRS:
Rules:
active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0')
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0')) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0')) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(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))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0') → ok(0')
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
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
isNat :: tt:mark:0':ok → tt:mark:0':ok
U13 :: 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
U31 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U32 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U33 :: tt:mark:0':ok → tt:mark:0':ok
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U51 :: 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
U61 :: tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
U71 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
x :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
isNatKind :: tt:mark:0':ok → 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)) → *4_0, rt ∈ Ω(n50)
U13(gen_tt:mark:0':ok3_0(+(1, n1763_0))) → *4_0, rt ∈ Ω(n17630)
U22(gen_tt:mark:0':ok3_0(+(1, n2483_0))) → *4_0, rt ∈ Ω(n24830)
U32(gen_tt:mark:0':ok3_0(+(1, n3304_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n33040)
U33(gen_tt:mark:0':ok3_0(+(1, n5769_0))) → *4_0, rt ∈ Ω(n57690)
s(gen_tt:mark:0':ok3_0(+(1, n6841_0))) → *4_0, rt ∈ Ω(n68410)
plus(gen_tt:mark:0':ok3_0(+(1, n8014_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n80140)
x(gen_tt:mark:0':ok3_0(+(1, n11394_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n113940)
U11(gen_tt:mark:0':ok3_0(+(1, n15078_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n150780)
and(gen_tt:mark:0':ok3_0(+(1, n21186_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n211860)
U21(gen_tt:mark:0':ok3_0(+(1, n25425_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n254250)
U31(gen_tt:mark:0':ok3_0(+(1, n29920_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299200)
U41(gen_tt:mark:0':ok3_0(+(1, n37549_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n375490)

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))

No more defined symbols left to analyse.

(65) Obligation:

TRS:
Rules:
active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0')
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0')) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0')) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) → mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2, X3)) → U31(active(X1), X2, X3)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(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))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2, X3) → mark(U31(X1, X2, X3))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(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))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2, X3)) → U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0') → ok(0')
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2), ok(X3)) → ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
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
isNat :: tt:mark:0':ok → tt:mark:0':ok
U13 :: 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
U31 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U32 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U33 :: tt:mark:0':ok → tt:mark:0':ok
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U51 :: 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
U61 :: tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
U71 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
x :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
isNatKind :: tt:mark:0':ok → 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)) → *4_0, rt ∈ Ω(n50)
U13(gen_tt:mark:0':ok3_0(+(1, n1763_0))) → *4_0, rt ∈ Ω(n17630)
U22(gen_tt:mark:0':ok3_0(+(1, n2483_0))) → *4_0, rt ∈ Ω(n24830)
U32(gen_tt:mark:0':ok3_0(+(1, n3304_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n33040)
U33(gen_tt:mark:0':ok3_0(+(1, n5769_0))) → *4_0, rt ∈ Ω(n57690)
s(gen_tt:mark:0':ok3_0(+(1, n6841_0))) → *4_0, rt ∈ Ω(n68410)
plus(gen_tt:mark:0':ok3_0(+(1, n8014_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n80140)
x(gen_tt:mark:0':ok3_0(+(1, n11394_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n113940)
U11(gen_tt:mark:0':ok3_0(+(1, n15078_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n150780)
and(gen_tt:mark:0':ok3_0(+(1, n21186_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n211860)
U21(gen_tt:mark:0':ok3_0(+(1, n25425_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n254250)
U31(gen_tt:mark:0':ok3_0(+(1, n29920_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n299200)

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))

No more defined symbols left to analyse.

(66) Obligation:

TRS:
Rules:
active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V1, V2)) → mark(U32(isNat(V1), V2))
active(U32(tt, V2)) → mark(U33(isNat(V2)))
active(U33(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0')
active(U71(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNat(x(V1, V2))) → mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNatKind(0')) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatKind(x(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(plus(N, 0')) → mark(U41(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(x(N, 0')) → mark(U61(and(isNat(N), isNatKind(N))))
active(x(N, s(M))) →