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

(0) Obligation:

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

U101(tt, M, N) → U102(isNatKind(activate(M)), activate(M), activate(N))
U102(tt, M, N) → U103(isNat(activate(N)), activate(M), activate(N))
U103(tt, M, N) → U104(isNatKind(activate(N)), activate(M), activate(N))
U104(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V1, V2) → U32(isNatKind(activate(V1)), activate(V1), activate(V2))
U32(tt, V1, V2) → U33(isNatKind(activate(V2)), activate(V1), activate(V2))
U33(tt, V1, V2) → U34(isNatKind(activate(V2)), activate(V1), activate(V2))
U34(tt, V1, V2) → U35(isNat(activate(V1)), activate(V2))
U35(tt, V2) → U36(isNat(activate(V2)))
U36(tt) → tt
U41(tt, V2) → U42(isNatKind(activate(V2)))
U42(tt) → tt
U51(tt) → tt
U61(tt, V2) → U62(isNatKind(activate(V2)))
U62(tt) → tt
U71(tt, N) → U72(isNatKind(activate(N)), activate(N))
U72(tt, N) → activate(N)
U81(tt, M, N) → U82(isNatKind(activate(M)), activate(M), activate(N))
U82(tt, M, N) → U83(isNat(activate(N)), activate(M), activate(N))
U83(tt, M, N) → U84(isNatKind(activate(N)), activate(M), activate(N))
U84(tt, M, N) → s(plus(activate(N), activate(M)))
U91(tt, N) → U92(isNatKind(activate(N)))
U92(tt) → 0
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__x(V1, V2)) → U31(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U41(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U51(isNatKind(activate(V1)))
isNatKind(n__x(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
plus(N, 0) → U71(isNat(N), N)
plus(N, s(M)) → U81(isNat(M), M, N)
x(N, 0) → U91(isNat(N), N)
x(N, s(M)) → U101(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(X2))
activate(X) → X

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(2ⁿ):
The rewrite sequence
activate(n__plus(X1, n__0)) →⁺ U71(isNat(activate(X1)), activate(X1))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
The pumping substitution is [X1 / n__plus(X1, n__0)].
The result substitution is [ ].

The rewrite sequence
activate(n__plus(X1, n__0)) →⁺ U71(isNat(activate(X1)), activate(X1))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
The pumping substitution is [X1 / n__plus(X1, n__0)].
The result substitution is [ ].

(2) BOUNDS(2^n, 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:

U101(tt, M, N) → U102(isNatKind(activate(M)), activate(M), activate(N))
U102(tt, M, N) → U103(isNat(activate(N)), activate(M), activate(N))
U103(tt, M, N) → U104(isNatKind(activate(N)), activate(M), activate(N))
U104(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V1, V2) → U32(isNatKind(activate(V1)), activate(V1), activate(V2))
U32(tt, V1, V2) → U33(isNatKind(activate(V2)), activate(V1), activate(V2))
U33(tt, V1, V2) → U34(isNatKind(activate(V2)), activate(V1), activate(V2))
U34(tt, V1, V2) → U35(isNat(activate(V1)), activate(V2))
U35(tt, V2) → U36(isNat(activate(V2)))
U36(tt) → tt
U41(tt, V2) → U42(isNatKind(activate(V2)))
U42(tt) → tt
U51(tt) → tt
U61(tt, V2) → U62(isNatKind(activate(V2)))
U62(tt) → tt
U71(tt, N) → U72(isNatKind(activate(N)), activate(N))
U72(tt, N) → activate(N)
U81(tt, M, N) → U82(isNatKind(activate(M)), activate(M), activate(N))
U82(tt, M, N) → U83(isNat(activate(N)), activate(M), activate(N))
U83(tt, M, N) → U84(isNatKind(activate(N)), activate(M), activate(N))
U84(tt, M, N) → s(plus(activate(N), activate(M)))
U91(tt, N) → U92(isNatKind(activate(N)))
U92(tt) → 0'
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__x(V1, V2)) → U31(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U41(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U51(isNatKind(activate(V1)))
isNatKind(n__x(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
plus(N, 0') → U71(isNat(N), N)
plus(N, s(M)) → U81(isNat(M), M, N)
x(N, 0') → U91(isNat(N), N)
x(N, s(M)) → U101(isNat(M), M, N)
0' → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0'
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(X2))
activate(X) → X

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
U101(tt, M, N) → U102(isNatKind(activate(M)), activate(M), activate(N))
U102(tt, M, N) → U103(isNat(activate(N)), activate(M), activate(N))
U103(tt, M, N) → U104(isNatKind(activate(N)), activate(M), activate(N))
U104(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V1, V2) → U32(isNatKind(activate(V1)), activate(V1), activate(V2))
U32(tt, V1, V2) → U33(isNatKind(activate(V2)), activate(V1), activate(V2))
U33(tt, V1, V2) → U34(isNatKind(activate(V2)), activate(V1), activate(V2))
U34(tt, V1, V2) → U35(isNat(activate(V1)), activate(V2))
U35(tt, V2) → U36(isNat(activate(V2)))
U36(tt) → tt
U41(tt, V2) → U42(isNatKind(activate(V2)))
U42(tt) → tt
U51(tt) → tt
U61(tt, V2) → U62(isNatKind(activate(V2)))
U62(tt) → tt
U71(tt, N) → U72(isNatKind(activate(N)), activate(N))
U72(tt, N) → activate(N)
U81(tt, M, N) → U82(isNatKind(activate(M)), activate(M), activate(N))
U82(tt, M, N) → U83(isNat(activate(N)), activate(M), activate(N))
U83(tt, M, N) → U84(isNatKind(activate(N)), activate(M), activate(N))
U84(tt, M, N) → s(plus(activate(N), activate(M)))
U91(tt, N) → U92(isNatKind(activate(N)))
U92(tt) → 0'
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__x(V1, V2)) → U31(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U41(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U51(isNatKind(activate(V1)))
isNatKind(n__x(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
plus(N, 0') → U71(isNat(N), N)
plus(N, s(M)) → U81(isNat(M), M, N)
x(N, 0') → U91(isNat(N), N)
x(N, s(M)) → U101(isNat(M), M, N)
0' → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0'
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(X2))
activate(X) → X

Types:
U101 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
tt :: tt
U102 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
isNatKind :: n__0:n__plus:n__s:n__x → tt
activate :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U103 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
isNat :: n__0:n__plus:n__s:n__x → tt
U104 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U11 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U12 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U13 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U14 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U15 :: tt → n__0:n__plus:n__s:n__x → tt
U16 :: tt → tt
U21 :: tt → n__0:n__plus:n__s:n__x → tt
U22 :: tt → n__0:n__plus:n__s:n__x → tt
U23 :: tt → tt
U31 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U32 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U33 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U34 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U35 :: tt → n__0:n__plus:n__s:n__x → tt
U36 :: tt → tt
U41 :: tt → n__0:n__plus:n__s:n__x → tt
U42 :: tt → tt
U51 :: tt → tt
U61 :: tt → n__0:n__plus:n__s:n__x → tt
U62 :: tt → tt
U71 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U72 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U81 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U82 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U83 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U84 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U91 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U92 :: tt → n__0:n__plus:n__s:n__x
0' :: n__0:n__plus:n__s:n__x
n__0 :: n__0:n__plus:n__s:n__x
n__plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
hole_n__0:n__plus:n__s:n__x1_7 :: n__0:n__plus:n__s:n__x
hole_tt2_7 :: tt
gen_n__0:n__plus:n__s:n__x3_7 :: Nat → n__0:n__plus:n__s:n__x

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

Heuristically decided to analyse the following defined symbols:
isNatKind, activate, isNat, plus, x, U71, U72, U91

They will be analysed ascendingly in the following order:
isNatKind = activate
isNatKind = isNat
isNatKind = plus
isNatKind = x
isNatKind = U71
isNatKind = U72
isNatKind = U91
activate = isNat
activate = plus
activate = x
activate = U71
activate = U72
activate = U91
isNat = plus
isNat = x
isNat = U71
isNat = U72
isNat = U91
plus = x
plus = U71
plus = U72
plus = U91
x = U71
x = U72
x = U91
U71 = U72
U71 = U91
U72 = U91

(8) Obligation:

TRS:
Rules:
U101(tt, M, N) → U102(isNatKind(activate(M)), activate(M), activate(N))
U102(tt, M, N) → U103(isNat(activate(N)), activate(M), activate(N))
U103(tt, M, N) → U104(isNatKind(activate(N)), activate(M), activate(N))
U104(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V1, V2) → U32(isNatKind(activate(V1)), activate(V1), activate(V2))
U32(tt, V1, V2) → U33(isNatKind(activate(V2)), activate(V1), activate(V2))
U33(tt, V1, V2) → U34(isNatKind(activate(V2)), activate(V1), activate(V2))
U34(tt, V1, V2) → U35(isNat(activate(V1)), activate(V2))
U35(tt, V2) → U36(isNat(activate(V2)))
U36(tt) → tt
U41(tt, V2) → U42(isNatKind(activate(V2)))
U42(tt) → tt
U51(tt) → tt
U61(tt, V2) → U62(isNatKind(activate(V2)))
U62(tt) → tt
U71(tt, N) → U72(isNatKind(activate(N)), activate(N))
U72(tt, N) → activate(N)
U81(tt, M, N) → U82(isNatKind(activate(M)), activate(M), activate(N))
U82(tt, M, N) → U83(isNat(activate(N)), activate(M), activate(N))
U83(tt, M, N) → U84(isNatKind(activate(N)), activate(M), activate(N))
U84(tt, M, N) → s(plus(activate(N), activate(M)))
U91(tt, N) → U92(isNatKind(activate(N)))
U92(tt) → 0'
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__x(V1, V2)) → U31(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U41(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U51(isNatKind(activate(V1)))
isNatKind(n__x(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
plus(N, 0') → U71(isNat(N), N)
plus(N, s(M)) → U81(isNat(M), M, N)
x(N, 0') → U91(isNat(N), N)
x(N, s(M)) → U101(isNat(M), M, N)
0' → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0'
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(X2))
activate(X) → X

Types:
U101 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
tt :: tt
U102 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
isNatKind :: n__0:n__plus:n__s:n__x → tt
activate :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U103 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
isNat :: n__0:n__plus:n__s:n__x → tt
U104 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U11 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U12 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U13 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U14 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U15 :: tt → n__0:n__plus:n__s:n__x → tt
U16 :: tt → tt
U21 :: tt → n__0:n__plus:n__s:n__x → tt
U22 :: tt → n__0:n__plus:n__s:n__x → tt
U23 :: tt → tt
U31 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U32 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U33 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U34 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U35 :: tt → n__0:n__plus:n__s:n__x → tt
U36 :: tt → tt
U41 :: tt → n__0:n__plus:n__s:n__x → tt
U42 :: tt → tt
U51 :: tt → tt
U61 :: tt → n__0:n__plus:n__s:n__x → tt
U62 :: tt → tt
U71 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U72 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U81 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U82 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U83 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U84 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U91 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U92 :: tt → n__0:n__plus:n__s:n__x
0' :: n__0:n__plus:n__s:n__x
n__0 :: n__0:n__plus:n__s:n__x
n__plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
hole_n__0:n__plus:n__s:n__x1_7 :: n__0:n__plus:n__s:n__x
hole_tt2_7 :: tt
gen_n__0:n__plus:n__s:n__x3_7 :: Nat → n__0:n__plus:n__s:n__x

Generator Equations:
gen_n__0:n__plus:n__s:n__x3_7(0) ⇔ n__0
gen_n__0:n__plus:n__s:n__x3_7(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s:n__x3_7(x), n__0)

The following defined symbols remain to be analysed:
activate, isNatKind, isNat, plus, x, U71, U72, U91

They will be analysed ascendingly in the following order:
isNatKind = activate
isNatKind = isNat
isNatKind = plus
isNatKind = x
isNatKind = U71
isNatKind = U72
isNatKind = U91
activate = isNat
activate = plus
activate = x
activate = U71
activate = U72
activate = U91
isNat = plus
isNat = x
isNat = U71
isNat = U72
isNat = U91
plus = x
plus = U71
plus = U72
plus = U91
x = U71
x = U72
x = U91
U71 = U72
U71 = U91
U72 = U91

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

Proved the following rewrite lemma:
activate(gen_n__0:n__plus:n__s:n__x3_7(n5_7)) → gen_n__0:n__plus:n__s:n__x3_7(n5_7), rt ∈ Ω(1 + n5₇)

Induction Base:
activate(gen_n__0:n__plus:n__s:n__x3_7(0)) →_R^Ω(1)
gen_n__0:n__plus:n__s:n__x3_7(0)

Induction Step:
activate(gen_n__0:n__plus:n__s:n__x3_7(+(n5_7, 1))) →_R^Ω(1)
plus(activate(gen_n__0:n__plus:n__s:n__x3_7(n5_7)), activate(n__0)) →_IH
plus(gen_n__0:n__plus:n__s:n__x3_7(c6_7), activate(n__0)) →_R^Ω(1)
plus(gen_n__0:n__plus:n__s:n__x3_7(n5_7), n__0) →_R^Ω(1)
n__plus(gen_n__0:n__plus:n__s:n__x3_7(n5_7), n__0)

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

(10) Complex Obligation (BEST)

(11) Obligation:

TRS:
Rules:
U101(tt, M, N) → U102(isNatKind(activate(M)), activate(M), activate(N))
U102(tt, M, N) → U103(isNat(activate(N)), activate(M), activate(N))
U103(tt, M, N) → U104(isNatKind(activate(N)), activate(M), activate(N))
U104(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V1, V2) → U32(isNatKind(activate(V1)), activate(V1), activate(V2))
U32(tt, V1, V2) → U33(isNatKind(activate(V2)), activate(V1), activate(V2))
U33(tt, V1, V2) → U34(isNatKind(activate(V2)), activate(V1), activate(V2))
U34(tt, V1, V2) → U35(isNat(activate(V1)), activate(V2))
U35(tt, V2) → U36(isNat(activate(V2)))
U36(tt) → tt
U41(tt, V2) → U42(isNatKind(activate(V2)))
U42(tt) → tt
U51(tt) → tt
U61(tt, V2) → U62(isNatKind(activate(V2)))
U62(tt) → tt
U71(tt, N) → U72(isNatKind(activate(N)), activate(N))
U72(tt, N) → activate(N)
U81(tt, M, N) → U82(isNatKind(activate(M)), activate(M), activate(N))
U82(tt, M, N) → U83(isNat(activate(N)), activate(M), activate(N))
U83(tt, M, N) → U84(isNatKind(activate(N)), activate(M), activate(N))
U84(tt, M, N) → s(plus(activate(N), activate(M)))
U91(tt, N) → U92(isNatKind(activate(N)))
U92(tt) → 0'
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__x(V1, V2)) → U31(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U41(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U51(isNatKind(activate(V1)))
isNatKind(n__x(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
plus(N, 0') → U71(isNat(N), N)
plus(N, s(M)) → U81(isNat(M), M, N)
x(N, 0') → U91(isNat(N), N)
x(N, s(M)) → U101(isNat(M), M, N)
0' → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0'
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(X2))
activate(X) → X

Types:
U101 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
tt :: tt
U102 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
isNatKind :: n__0:n__plus:n__s:n__x → tt
activate :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U103 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
isNat :: n__0:n__plus:n__s:n__x → tt
U104 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U11 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U12 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U13 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U14 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U15 :: tt → n__0:n__plus:n__s:n__x → tt
U16 :: tt → tt
U21 :: tt → n__0:n__plus:n__s:n__x → tt
U22 :: tt → n__0:n__plus:n__s:n__x → tt
U23 :: tt → tt
U31 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U32 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U33 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U34 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U35 :: tt → n__0:n__plus:n__s:n__x → tt
U36 :: tt → tt
U41 :: tt → n__0:n__plus:n__s:n__x → tt
U42 :: tt → tt
U51 :: tt → tt
U61 :: tt → n__0:n__plus:n__s:n__x → tt
U62 :: tt → tt
U71 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U72 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U81 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U82 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U83 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U84 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U91 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U92 :: tt → n__0:n__plus:n__s:n__x
0' :: n__0:n__plus:n__s:n__x
n__0 :: n__0:n__plus:n__s:n__x
n__plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
hole_n__0:n__plus:n__s:n__x1_7 :: n__0:n__plus:n__s:n__x
hole_tt2_7 :: tt
gen_n__0:n__plus:n__s:n__x3_7 :: Nat → n__0:n__plus:n__s:n__x

Lemmas:
activate(gen_n__0:n__plus:n__s:n__x3_7(n5_7)) → gen_n__0:n__plus:n__s:n__x3_7(n5_7), rt ∈ Ω(1 + n5₇)

Generator Equations:
gen_n__0:n__plus:n__s:n__x3_7(0) ⇔ n__0
gen_n__0:n__plus:n__s:n__x3_7(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s:n__x3_7(x), n__0)

The following defined symbols remain to be analysed:
plus, isNatKind, isNat, x, U71, U72, U91

They will be analysed ascendingly in the following order:
isNatKind = activate
isNatKind = isNat
isNatKind = plus
isNatKind = x
isNatKind = U71
isNatKind = U72
isNatKind = U91
activate = isNat
activate = plus
activate = x
activate = U71
activate = U72
activate = U91
isNat = plus
isNat = x
isNat = U71
isNat = U72
isNat = U91
plus = x
plus = U71
plus = U72
plus = U91
x = U71
x = U72
x = U91
U71 = U72
U71 = U91
U72 = U91

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

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

(13) Obligation:

TRS:
Rules:
U101(tt, M, N) → U102(isNatKind(activate(M)), activate(M), activate(N))
U102(tt, M, N) → U103(isNat(activate(N)), activate(M), activate(N))
U103(tt, M, N) → U104(isNatKind(activate(N)), activate(M), activate(N))
U104(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V1, V2) → U32(isNatKind(activate(V1)), activate(V1), activate(V2))
U32(tt, V1, V2) → U33(isNatKind(activate(V2)), activate(V1), activate(V2))
U33(tt, V1, V2) → U34(isNatKind(activate(V2)), activate(V1), activate(V2))
U34(tt, V1, V2) → U35(isNat(activate(V1)), activate(V2))
U35(tt, V2) → U36(isNat(activate(V2)))
U36(tt) → tt
U41(tt, V2) → U42(isNatKind(activate(V2)))
U42(tt) → tt
U51(tt) → tt
U61(tt, V2) → U62(isNatKind(activate(V2)))
U62(tt) → tt
U71(tt, N) → U72(isNatKind(activate(N)), activate(N))
U72(tt, N) → activate(N)
U81(tt, M, N) → U82(isNatKind(activate(M)), activate(M), activate(N))
U82(tt, M, N) → U83(isNat(activate(N)), activate(M), activate(N))
U83(tt, M, N) → U84(isNatKind(activate(N)), activate(M), activate(N))
U84(tt, M, N) → s(plus(activate(N), activate(M)))
U91(tt, N) → U92(isNatKind(activate(N)))
U92(tt) → 0'
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__x(V1, V2)) → U31(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U41(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U51(isNatKind(activate(V1)))
isNatKind(n__x(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
plus(N, 0') → U71(isNat(N), N)
plus(N, s(M)) → U81(isNat(M), M, N)
x(N, 0') → U91(isNat(N), N)
x(N, s(M)) → U101(isNat(M), M, N)
0' → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0'
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(X2))
activate(X) → X

Types:
U101 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
tt :: tt
U102 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
isNatKind :: n__0:n__plus:n__s:n__x → tt
activate :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U103 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
isNat :: n__0:n__plus:n__s:n__x → tt
U104 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U11 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U12 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U13 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U14 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U15 :: tt → n__0:n__plus:n__s:n__x → tt
U16 :: tt → tt
U21 :: tt → n__0:n__plus:n__s:n__x → tt
U22 :: tt → n__0:n__plus:n__s:n__x → tt
U23 :: tt → tt
U31 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U32 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U33 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U34 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U35 :: tt → n__0:n__plus:n__s:n__x → tt
U36 :: tt → tt
U41 :: tt → n__0:n__plus:n__s:n__x → tt
U42 :: tt → tt
U51 :: tt → tt
U61 :: tt → n__0:n__plus:n__s:n__x → tt
U62 :: tt → tt
U71 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U72 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U81 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U82 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U83 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U84 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U91 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U92 :: tt → n__0:n__plus:n__s:n__x
0' :: n__0:n__plus:n__s:n__x
n__0 :: n__0:n__plus:n__s:n__x
n__plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
hole_n__0:n__plus:n__s:n__x1_7 :: n__0:n__plus:n__s:n__x
hole_tt2_7 :: tt
gen_n__0:n__plus:n__s:n__x3_7 :: Nat → n__0:n__plus:n__s:n__x

Lemmas:
activate(gen_n__0:n__plus:n__s:n__x3_7(n5_7)) → gen_n__0:n__plus:n__s:n__x3_7(n5_7), rt ∈ Ω(1 + n5₇)

Generator Equations:
gen_n__0:n__plus:n__s:n__x3_7(0) ⇔ n__0
gen_n__0:n__plus:n__s:n__x3_7(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s:n__x3_7(x), n__0)

The following defined symbols remain to be analysed:
U71, isNatKind, isNat, x, U72, U91

They will be analysed ascendingly in the following order:
isNatKind = activate
isNatKind = isNat
isNatKind = plus
isNatKind = x
isNatKind = U71
isNatKind = U72
isNatKind = U91
activate = isNat
activate = plus
activate = x
activate = U71
activate = U72
activate = U91
isNat = plus
isNat = x
isNat = U71
isNat = U72
isNat = U91
plus = x
plus = U71
plus = U72
plus = U91
x = U71
x = U72
x = U91
U71 = U72
U71 = U91
U72 = U91

(14) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U71.

(15) Obligation:

TRS:
Rules:
U101(tt, M, N) → U102(isNatKind(activate(M)), activate(M), activate(N))
U102(tt, M, N) → U103(isNat(activate(N)), activate(M), activate(N))
U103(tt, M, N) → U104(isNatKind(activate(N)), activate(M), activate(N))
U104(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V1, V2) → U32(isNatKind(activate(V1)), activate(V1), activate(V2))
U32(tt, V1, V2) → U33(isNatKind(activate(V2)), activate(V1), activate(V2))
U33(tt, V1, V2) → U34(isNatKind(activate(V2)), activate(V1), activate(V2))
U34(tt, V1, V2) → U35(isNat(activate(V1)), activate(V2))
U35(tt, V2) → U36(isNat(activate(V2)))
U36(tt) → tt
U41(tt, V2) → U42(isNatKind(activate(V2)))
U42(tt) → tt
U51(tt) → tt
U61(tt, V2) → U62(isNatKind(activate(V2)))
U62(tt) → tt
U71(tt, N) → U72(isNatKind(activate(N)), activate(N))
U72(tt, N) → activate(N)
U81(tt, M, N) → U82(isNatKind(activate(M)), activate(M), activate(N))
U82(tt, M, N) → U83(isNat(activate(N)), activate(M), activate(N))
U83(tt, M, N) → U84(isNatKind(activate(N)), activate(M), activate(N))
U84(tt, M, N) → s(plus(activate(N), activate(M)))
U91(tt, N) → U92(isNatKind(activate(N)))
U92(tt) → 0'
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__x(V1, V2)) → U31(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U41(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U51(isNatKind(activate(V1)))
isNatKind(n__x(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
plus(N, 0') → U71(isNat(N), N)
plus(N, s(M)) → U81(isNat(M), M, N)
x(N, 0') → U91(isNat(N), N)
x(N, s(M)) → U101(isNat(M), M, N)
0' → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0'
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(X2))
activate(X) → X

Types:
U101 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
tt :: tt
U102 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
isNatKind :: n__0:n__plus:n__s:n__x → tt
activate :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U103 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
isNat :: n__0:n__plus:n__s:n__x → tt
U104 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U11 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U12 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U13 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U14 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U15 :: tt → n__0:n__plus:n__s:n__x → tt
U16 :: tt → tt
U21 :: tt → n__0:n__plus:n__s:n__x → tt
U22 :: tt → n__0:n__plus:n__s:n__x → tt
U23 :: tt → tt
U31 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U32 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U33 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U34 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U35 :: tt → n__0:n__plus:n__s:n__x → tt
U36 :: tt → tt
U41 :: tt → n__0:n__plus:n__s:n__x → tt
U42 :: tt → tt
U51 :: tt → tt
U61 :: tt → n__0:n__plus:n__s:n__x → tt
U62 :: tt → tt
U71 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U72 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U81 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U82 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U83 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U84 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U91 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U92 :: tt → n__0:n__plus:n__s:n__x
0' :: n__0:n__plus:n__s:n__x
n__0 :: n__0:n__plus:n__s:n__x
n__plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
hole_n__0:n__plus:n__s:n__x1_7 :: n__0:n__plus:n__s:n__x
hole_tt2_7 :: tt
gen_n__0:n__plus:n__s:n__x3_7 :: Nat → n__0:n__plus:n__s:n__x

Lemmas:
activate(gen_n__0:n__plus:n__s:n__x3_7(n5_7)) → gen_n__0:n__plus:n__s:n__x3_7(n5_7), rt ∈ Ω(1 + n5₇)

Generator Equations:
gen_n__0:n__plus:n__s:n__x3_7(0) ⇔ n__0
gen_n__0:n__plus:n__s:n__x3_7(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s:n__x3_7(x), n__0)

The following defined symbols remain to be analysed:
U72, isNatKind, isNat, x, U91

They will be analysed ascendingly in the following order:
isNatKind = activate
isNatKind = isNat
isNatKind = plus
isNatKind = x
isNatKind = U71
isNatKind = U72
isNatKind = U91
activate = isNat
activate = plus
activate = x
activate = U71
activate = U72
activate = U91
isNat = plus
isNat = x
isNat = U71
isNat = U72
isNat = U91
plus = x
plus = U71
plus = U72
plus = U91
x = U71
x = U72
x = U91
U71 = U72
U71 = U91
U72 = U91

(16) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U72.

(17) Obligation:

TRS:
Rules:
U101(tt, M, N) → U102(isNatKind(activate(M)), activate(M), activate(N))
U102(tt, M, N) → U103(isNat(activate(N)), activate(M), activate(N))
U103(tt, M, N) → U104(isNatKind(activate(N)), activate(M), activate(N))
U104(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V1, V2) → U32(isNatKind(activate(V1)), activate(V1), activate(V2))
U32(tt, V1, V2) → U33(isNatKind(activate(V2)), activate(V1), activate(V2))
U33(tt, V1, V2) → U34(isNatKind(activate(V2)), activate(V1), activate(V2))
U34(tt, V1, V2) → U35(isNat(activate(V1)), activate(V2))
U35(tt, V2) → U36(isNat(activate(V2)))
U36(tt) → tt
U41(tt, V2) → U42(isNatKind(activate(V2)))
U42(tt) → tt
U51(tt) → tt
U61(tt, V2) → U62(isNatKind(activate(V2)))
U62(tt) → tt
U71(tt, N) → U72(isNatKind(activate(N)), activate(N))
U72(tt, N) → activate(N)
U81(tt, M, N) → U82(isNatKind(activate(M)), activate(M), activate(N))
U82(tt, M, N) → U83(isNat(activate(N)), activate(M), activate(N))
U83(tt, M, N) → U84(isNatKind(activate(N)), activate(M), activate(N))
U84(tt, M, N) → s(plus(activate(N), activate(M)))
U91(tt, N) → U92(isNatKind(activate(N)))
U92(tt) → 0'
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__x(V1, V2)) → U31(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U41(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U51(isNatKind(activate(V1)))
isNatKind(n__x(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
plus(N, 0') → U71(isNat(N), N)
plus(N, s(M)) → U81(isNat(M), M, N)
x(N, 0') → U91(isNat(N), N)
x(N, s(M)) → U101(isNat(M), M, N)
0' → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0'
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(X2))
activate(X) → X

Types:
U101 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
tt :: tt
U102 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
isNatKind :: n__0:n__plus:n__s:n__x → tt
activate :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U103 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
isNat :: n__0:n__plus:n__s:n__x → tt
U104 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U11 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U12 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U13 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U14 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U15 :: tt → n__0:n__plus:n__s:n__x → tt
U16 :: tt → tt
U21 :: tt → n__0:n__plus:n__s:n__x → tt
U22 :: tt → n__0:n__plus:n__s:n__x → tt
U23 :: tt → tt
U31 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U32 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U33 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U34 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U35 :: tt → n__0:n__plus:n__s:n__x → tt
U36 :: tt → tt
U41 :: tt → n__0:n__plus:n__s:n__x → tt
U42 :: tt → tt
U51 :: tt → tt
U61 :: tt → n__0:n__plus:n__s:n__x → tt
U62 :: tt → tt
U71 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U72 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U81 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U82 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U83 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U84 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U91 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U92 :: tt → n__0:n__plus:n__s:n__x
0' :: n__0:n__plus:n__s:n__x
n__0 :: n__0:n__plus:n__s:n__x
n__plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
hole_n__0:n__plus:n__s:n__x1_7 :: n__0:n__plus:n__s:n__x
hole_tt2_7 :: tt
gen_n__0:n__plus:n__s:n__x3_7 :: Nat → n__0:n__plus:n__s:n__x

Lemmas:
activate(gen_n__0:n__plus:n__s:n__x3_7(n5_7)) → gen_n__0:n__plus:n__s:n__x3_7(n5_7), rt ∈ Ω(1 + n5₇)

Generator Equations:
gen_n__0:n__plus:n__s:n__x3_7(0) ⇔ n__0
gen_n__0:n__plus:n__s:n__x3_7(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s:n__x3_7(x), n__0)

The following defined symbols remain to be analysed:
isNatKind, isNat, x, U91

They will be analysed ascendingly in the following order:
isNatKind = activate
isNatKind = isNat
isNatKind = plus
isNatKind = x
isNatKind = U71
isNatKind = U72
isNatKind = U91
activate = isNat
activate = plus
activate = x
activate = U71
activate = U72
activate = U91
isNat = plus
isNat = x
isNat = U71
isNat = U72
isNat = U91
plus = x
plus = U71
plus = U72
plus = U91
x = U71
x = U72
x = U91
U71 = U72
U71 = U91
U72 = U91

(18) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
isNatKind(gen_n__0:n__plus:n__s:n__x3_7(n14928_7)) → tt, rt ∈ Ω(1 + n14928₇ + n14928₇²)

Induction Base:
isNatKind(gen_n__0:n__plus:n__s:n__x3_7(0)) →_R^Ω(1)
tt

Induction Step:
isNatKind(gen_n__0:n__plus:n__s:n__x3_7(+(n14928_7, 1))) →_R^Ω(1)
U41(isNatKind(activate(gen_n__0:n__plus:n__s:n__x3_7(n14928_7))), activate(n__0)) →_L^{Ω(1 + n14928₇)}
U41(isNatKind(gen_n__0:n__plus:n__s:n__x3_7(n14928_7)), activate(n__0)) →_IH
U41(tt, activate(n__0)) →_L^Ω(1)
U41(tt, gen_n__0:n__plus:n__s:n__x3_7(0)) →_R^Ω(1)
U42(isNatKind(activate(gen_n__0:n__plus:n__s:n__x3_7(0)))) →_L^Ω(1)
U42(isNatKind(gen_n__0:n__plus:n__s:n__x3_7(0))) →_R^Ω(1)
U42(tt) →_R^Ω(1)
tt

We have rt ∈ Ω(n²) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n²).

(19) Complex Obligation (BEST)

(20) Obligation:

TRS:
Rules:
U101(tt, M, N) → U102(isNatKind(activate(M)), activate(M), activate(N))
U102(tt, M, N) → U103(isNat(activate(N)), activate(M), activate(N))
U103(tt, M, N) → U104(isNatKind(activate(N)), activate(M), activate(N))
U104(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V1, V2) → U32(isNatKind(activate(V1)), activate(V1), activate(V2))
U32(tt, V1, V2) → U33(isNatKind(activate(V2)), activate(V1), activate(V2))
U33(tt, V1, V2) → U34(isNatKind(activate(V2)), activate(V1), activate(V2))
U34(tt, V1, V2) → U35(isNat(activate(V1)), activate(V2))
U35(tt, V2) → U36(isNat(activate(V2)))
U36(tt) → tt
U41(tt, V2) → U42(isNatKind(activate(V2)))
U42(tt) → tt
U51(tt) → tt
U61(tt, V2) → U62(isNatKind(activate(V2)))
U62(tt) → tt
U71(tt, N) → U72(isNatKind(activate(N)), activate(N))
U72(tt, N) → activate(N)
U81(tt, M, N) → U82(isNatKind(activate(M)), activate(M), activate(N))
U82(tt, M, N) → U83(isNat(activate(N)), activate(M), activate(N))
U83(tt, M, N) → U84(isNatKind(activate(N)), activate(M), activate(N))
U84(tt, M, N) → s(plus(activate(N), activate(M)))
U91(tt, N) → U92(isNatKind(activate(N)))
U92(tt) → 0'
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__x(V1, V2)) → U31(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U41(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U51(isNatKind(activate(V1)))
isNatKind(n__x(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
plus(N, 0') → U71(isNat(N), N)
plus(N, s(M)) → U81(isNat(M), M, N)
x(N, 0') → U91(isNat(N), N)
x(N, s(M)) → U101(isNat(M), M, N)
0' → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0'
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(X2))
activate(X) → X

Types:
U101 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
tt :: tt
U102 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
isNatKind :: n__0:n__plus:n__s:n__x → tt
activate :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U103 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
isNat :: n__0:n__plus:n__s:n__x → tt
U104 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U11 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U12 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U13 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U14 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U15 :: tt → n__0:n__plus:n__s:n__x → tt
U16 :: tt → tt
U21 :: tt → n__0:n__plus:n__s:n__x → tt
U22 :: tt → n__0:n__plus:n__s:n__x → tt
U23 :: tt → tt
U31 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U32 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U33 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U34 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U35 :: tt → n__0:n__plus:n__s:n__x → tt
U36 :: tt → tt
U41 :: tt → n__0:n__plus:n__s:n__x → tt
U42 :: tt → tt
U51 :: tt → tt
U61 :: tt → n__0:n__plus:n__s:n__x → tt
U62 :: tt → tt
U71 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U72 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U81 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U82 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U83 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U84 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U91 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U92 :: tt → n__0:n__plus:n__s:n__x
0' :: n__0:n__plus:n__s:n__x
n__0 :: n__0:n__plus:n__s:n__x
n__plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
hole_n__0:n__plus:n__s:n__x1_7 :: n__0:n__plus:n__s:n__x
hole_tt2_7 :: tt
gen_n__0:n__plus:n__s:n__x3_7 :: Nat → n__0:n__plus:n__s:n__x

Lemmas:
activate(gen_n__0:n__plus:n__s:n__x3_7(n5_7)) → gen_n__0:n__plus:n__s:n__x3_7(n5_7), rt ∈ Ω(1 + n5₇)
isNatKind(gen_n__0:n__plus:n__s:n__x3_7(n14928_7)) → tt, rt ∈ Ω(1 + n14928₇ + n14928₇²)

Generator Equations:
gen_n__0:n__plus:n__s:n__x3_7(0) ⇔ n__0
gen_n__0:n__plus:n__s:n__x3_7(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s:n__x3_7(x), n__0)

The following defined symbols remain to be analysed:
isNat, activate, plus, x, U71, U72, U91

They will be analysed ascendingly in the following order:
isNatKind = activate
isNatKind = isNat
isNatKind = plus
isNatKind = x
isNatKind = U71
isNatKind = U72
isNatKind = U91
activate = isNat
activate = plus
activate = x
activate = U71
activate = U72
activate = U91
isNat = plus
isNat = x
isNat = U71
isNat = U72
isNat = U91
plus = x
plus = U71
plus = U72
plus = U91
x = U71
x = U72
x = U91
U71 = U72
U71 = U91
U72 = U91

(21) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(22) Obligation:

TRS:
Rules:
U101(tt, M, N) → U102(isNatKind(activate(M)), activate(M), activate(N))
U102(tt, M, N) → U103(isNat(activate(N)), activate(M), activate(N))
U103(tt, M, N) → U104(isNatKind(activate(N)), activate(M), activate(N))
U104(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V1, V2) → U32(isNatKind(activate(V1)), activate(V1), activate(V2))
U32(tt, V1, V2) → U33(isNatKind(activate(V2)), activate(V1), activate(V2))
U33(tt, V1, V2) → U34(isNatKind(activate(V2)), activate(V1), activate(V2))
U34(tt, V1, V2) → U35(isNat(activate(V1)), activate(V2))
U35(tt, V2) → U36(isNat(activate(V2)))
U36(tt) → tt
U41(tt, V2) → U42(isNatKind(activate(V2)))
U42(tt) → tt
U51(tt) → tt
U61(tt, V2) → U62(isNatKind(activate(V2)))
U62(tt) → tt
U71(tt, N) → U72(isNatKind(activate(N)), activate(N))
U72(tt, N) → activate(N)
U81(tt, M, N) → U82(isNatKind(activate(M)), activate(M), activate(N))
U82(tt, M, N) → U83(isNat(activate(N)), activate(M), activate(N))
U83(tt, M, N) → U84(isNatKind(activate(N)), activate(M), activate(N))
U84(tt, M, N) → s(plus(activate(N), activate(M)))
U91(tt, N) → U92(isNatKind(activate(N)))
U92(tt) → 0'
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__x(V1, V2)) → U31(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U41(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U51(isNatKind(activate(V1)))
isNatKind(n__x(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
plus(N, 0') → U71(isNat(N), N)
plus(N, s(M)) → U81(isNat(M), M, N)
x(N, 0') → U91(isNat(N), N)
x(N, s(M)) → U101(isNat(M), M, N)
0' → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0'
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(X2))
activate(X) → X

Types:
U101 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
tt :: tt
U102 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
isNatKind :: n__0:n__plus:n__s:n__x → tt
activate :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U103 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
isNat :: n__0:n__plus:n__s:n__x → tt
U104 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U11 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U12 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U13 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U14 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U15 :: tt → n__0:n__plus:n__s:n__x → tt
U16 :: tt → tt
U21 :: tt → n__0:n__plus:n__s:n__x → tt
U22 :: tt → n__0:n__plus:n__s:n__x → tt
U23 :: tt → tt
U31 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U32 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U33 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U34 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U35 :: tt → n__0:n__plus:n__s:n__x → tt
U36 :: tt → tt
U41 :: tt → n__0:n__plus:n__s:n__x → tt
U42 :: tt → tt
U51 :: tt → tt
U61 :: tt → n__0:n__plus:n__s:n__x → tt
U62 :: tt → tt
U71 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U72 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U81 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U82 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U83 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U84 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U91 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U92 :: tt → n__0:n__plus:n__s:n__x
0' :: n__0:n__plus:n__s:n__x
n__0 :: n__0:n__plus:n__s:n__x
n__plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
hole_n__0:n__plus:n__s:n__x1_7 :: n__0:n__plus:n__s:n__x
hole_tt2_7 :: tt
gen_n__0:n__plus:n__s:n__x3_7 :: Nat → n__0:n__plus:n__s:n__x

Lemmas:
activate(gen_n__0:n__plus:n__s:n__x3_7(n5_7)) → gen_n__0:n__plus:n__s:n__x3_7(n5_7), rt ∈ Ω(1 + n5₇)
isNatKind(gen_n__0:n__plus:n__s:n__x3_7(n14928_7)) → tt, rt ∈ Ω(1 + n14928₇ + n14928₇²)

Generator Equations:
gen_n__0:n__plus:n__s:n__x3_7(0) ⇔ n__0
gen_n__0:n__plus:n__s:n__x3_7(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s:n__x3_7(x), n__0)

The following defined symbols remain to be analysed:
x, activate, plus, U71, U72, U91

They will be analysed ascendingly in the following order:
isNatKind = activate
isNatKind = isNat
isNatKind = plus
isNatKind = x
isNatKind = U71
isNatKind = U72
isNatKind = U91
activate = isNat
activate = plus
activate = x
activate = U71
activate = U72
activate = U91
isNat = plus
isNat = x
isNat = U71
isNat = U72
isNat = U91
plus = x
plus = U71
plus = U72
plus = U91
x = U71
x = U72
x = U91
U71 = U72
U71 = U91
U72 = U91

(23) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(24) Obligation:

TRS:
Rules:
U101(tt, M, N) → U102(isNatKind(activate(M)), activate(M), activate(N))
U102(tt, M, N) → U103(isNat(activate(N)), activate(M), activate(N))
U103(tt, M, N) → U104(isNatKind(activate(N)), activate(M), activate(N))
U104(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V1, V2) → U32(isNatKind(activate(V1)), activate(V1), activate(V2))
U32(tt, V1, V2) → U33(isNatKind(activate(V2)), activate(V1), activate(V2))
U33(tt, V1, V2) → U34(isNatKind(activate(V2)), activate(V1), activate(V2))
U34(tt, V1, V2) → U35(isNat(activate(V1)), activate(V2))
U35(tt, V2) → U36(isNat(activate(V2)))
U36(tt) → tt
U41(tt, V2) → U42(isNatKind(activate(V2)))
U42(tt) → tt
U51(tt) → tt
U61(tt, V2) → U62(isNatKind(activate(V2)))
U62(tt) → tt
U71(tt, N) → U72(isNatKind(activate(N)), activate(N))
U72(tt, N) → activate(N)
U81(tt, M, N) → U82(isNatKind(activate(M)), activate(M), activate(N))
U82(tt, M, N) → U83(isNat(activate(N)), activate(M), activate(N))
U83(tt, M, N) → U84(isNatKind(activate(N)), activate(M), activate(N))
U84(tt, M, N) → s(plus(activate(N), activate(M)))
U91(tt, N) → U92(isNatKind(activate(N)))
U92(tt) → 0'
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__x(V1, V2)) → U31(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U41(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U51(isNatKind(activate(V1)))
isNatKind(n__x(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
plus(N, 0') → U71(isNat(N), N)
plus(N, s(M)) → U81(isNat(M), M, N)
x(N, 0') → U91(isNat(N), N)
x(N, s(M)) → U101(isNat(M), M, N)
0' → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0'
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(X2))
activate(X) → X

Types:
U101 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
tt :: tt
U102 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
isNatKind :: n__0:n__plus:n__s:n__x → tt
activate :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U103 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
isNat :: n__0:n__plus:n__s:n__x → tt
U104 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U11 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U12 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U13 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U14 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U15 :: tt → n__0:n__plus:n__s:n__x → tt
U16 :: tt → tt
U21 :: tt → n__0:n__plus:n__s:n__x → tt
U22 :: tt → n__0:n__plus:n__s:n__x → tt
U23 :: tt → tt
U31 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U32 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U33 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U34 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U35 :: tt → n__0:n__plus:n__s:n__x → tt
U36 :: tt → tt
U41 :: tt → n__0:n__plus:n__s:n__x → tt
U42 :: tt → tt
U51 :: tt → tt
U61 :: tt → n__0:n__plus:n__s:n__x → tt
U62 :: tt → tt
U71 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U72 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U81 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U82 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U83 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U84 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U91 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U92 :: tt → n__0:n__plus:n__s:n__x
0' :: n__0:n__plus:n__s:n__x
n__0 :: n__0:n__plus:n__s:n__x
n__plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
hole_n__0:n__plus:n__s:n__x1_7 :: n__0:n__plus:n__s:n__x
hole_tt2_7 :: tt
gen_n__0:n__plus:n__s:n__x3_7 :: Nat → n__0:n__plus:n__s:n__x

Lemmas:
activate(gen_n__0:n__plus:n__s:n__x3_7(n5_7)) → gen_n__0:n__plus:n__s:n__x3_7(n5_7), rt ∈ Ω(1 + n5₇)
isNatKind(gen_n__0:n__plus:n__s:n__x3_7(n14928_7)) → tt, rt ∈ Ω(1 + n14928₇ + n14928₇²)

Generator Equations:
gen_n__0:n__plus:n__s:n__x3_7(0) ⇔ n__0
gen_n__0:n__plus:n__s:n__x3_7(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s:n__x3_7(x), n__0)

The following defined symbols remain to be analysed:
U91, activate, plus, U71, U72

They will be analysed ascendingly in the following order:
isNatKind = activate
isNatKind = isNat
isNatKind = plus
isNatKind = x
isNatKind = U71
isNatKind = U72
isNatKind = U91
activate = isNat
activate = plus
activate = x
activate = U71
activate = U72
activate = U91
isNat = plus
isNat = x
isNat = U71
isNat = U72
isNat = U91
plus = x
plus = U71
plus = U72
plus = U91
x = U71
x = U72
x = U91
U71 = U72
U71 = U91
U72 = U91

(25) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U91.

(26) Obligation:

TRS:
Rules:
U101(tt, M, N) → U102(isNatKind(activate(M)), activate(M), activate(N))
U102(tt, M, N) → U103(isNat(activate(N)), activate(M), activate(N))
U103(tt, M, N) → U104(isNatKind(activate(N)), activate(M), activate(N))
U104(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V1, V2) → U32(isNatKind(activate(V1)), activate(V1), activate(V2))
U32(tt, V1, V2) → U33(isNatKind(activate(V2)), activate(V1), activate(V2))
U33(tt, V1, V2) → U34(isNatKind(activate(V2)), activate(V1), activate(V2))
U34(tt, V1, V2) → U35(isNat(activate(V1)), activate(V2))
U35(tt, V2) → U36(isNat(activate(V2)))
U36(tt) → tt
U41(tt, V2) → U42(isNatKind(activate(V2)))
U42(tt) → tt
U51(tt) → tt
U61(tt, V2) → U62(isNatKind(activate(V2)))
U62(tt) → tt
U71(tt, N) → U72(isNatKind(activate(N)), activate(N))
U72(tt, N) → activate(N)
U81(tt, M, N) → U82(isNatKind(activate(M)), activate(M), activate(N))
U82(tt, M, N) → U83(isNat(activate(N)), activate(M), activate(N))
U83(tt, M, N) → U84(isNatKind(activate(N)), activate(M), activate(N))
U84(tt, M, N) → s(plus(activate(N), activate(M)))
U91(tt, N) → U92(isNatKind(activate(N)))
U92(tt) → 0'
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__x(V1, V2)) → U31(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U41(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U51(isNatKind(activate(V1)))
isNatKind(n__x(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
plus(N, 0') → U71(isNat(N), N)
plus(N, s(M)) → U81(isNat(M), M, N)
x(N, 0') → U91(isNat(N), N)
x(N, s(M)) → U101(isNat(M), M, N)
0' → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0'
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(X2))
activate(X) → X

Types:
U101 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
tt :: tt
U102 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
isNatKind :: n__0:n__plus:n__s:n__x → tt
activate :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U103 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
isNat :: n__0:n__plus:n__s:n__x → tt
U104 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U11 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U12 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U13 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U14 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U15 :: tt → n__0:n__plus:n__s:n__x → tt
U16 :: tt → tt
U21 :: tt → n__0:n__plus:n__s:n__x → tt
U22 :: tt → n__0:n__plus:n__s:n__x → tt
U23 :: tt → tt
U31 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U32 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U33 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U34 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U35 :: tt → n__0:n__plus:n__s:n__x → tt
U36 :: tt → tt
U41 :: tt → n__0:n__plus:n__s:n__x → tt
U42 :: tt → tt
U51 :: tt → tt
U61 :: tt → n__0:n__plus:n__s:n__x → tt
U62 :: tt → tt
U71 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U72 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U81 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U82 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U83 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U84 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U91 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U92 :: tt → n__0:n__plus:n__s:n__x
0' :: n__0:n__plus:n__s:n__x
n__0 :: n__0:n__plus:n__s:n__x
n__plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
hole_n__0:n__plus:n__s:n__x1_7 :: n__0:n__plus:n__s:n__x
hole_tt2_7 :: tt
gen_n__0:n__plus:n__s:n__x3_7 :: Nat → n__0:n__plus:n__s:n__x

Lemmas:
activate(gen_n__0:n__plus:n__s:n__x3_7(n5_7)) → gen_n__0:n__plus:n__s:n__x3_7(n5_7), rt ∈ Ω(1 + n5₇)
isNatKind(gen_n__0:n__plus:n__s:n__x3_7(n14928_7)) → tt, rt ∈ Ω(1 + n14928₇ + n14928₇²)

Generator Equations:
gen_n__0:n__plus:n__s:n__x3_7(0) ⇔ n__0
gen_n__0:n__plus:n__s:n__x3_7(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s:n__x3_7(x), n__0)

The following defined symbols remain to be analysed:
activate, plus, U71, U72

They will be analysed ascendingly in the following order:
isNatKind = activate
isNatKind = isNat
isNatKind = plus
isNatKind = x
isNatKind = U71
isNatKind = U72
isNatKind = U91
activate = isNat
activate = plus
activate = x
activate = U71
activate = U72
activate = U91
isNat = plus
isNat = x
isNat = U71
isNat = U72
isNat = U91
plus = x
plus = U71
plus = U72
plus = U91
x = U71
x = U72
x = U91
U71 = U72
U71 = U91
U72 = U91

(27) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
activate(gen_n__0:n__plus:n__s:n__x3_7(n18510_7)) → gen_n__0:n__plus:n__s:n__x3_7(n18510_7), rt ∈ Ω(1 + n18510₇)

Induction Base:
activate(gen_n__0:n__plus:n__s:n__x3_7(0)) →_R^Ω(1)
gen_n__0:n__plus:n__s:n__x3_7(0)

Induction Step:
activate(gen_n__0:n__plus:n__s:n__x3_7(+(n18510_7, 1))) →_R^Ω(1)
plus(activate(gen_n__0:n__plus:n__s:n__x3_7(n18510_7)), activate(n__0)) →_IH
plus(gen_n__0:n__plus:n__s:n__x3_7(c18511_7), activate(n__0)) →_R^Ω(1)
plus(gen_n__0:n__plus:n__s:n__x3_7(n18510_7), n__0) →_R^Ω(1)
n__plus(gen_n__0:n__plus:n__s:n__x3_7(n18510_7), n__0)

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

(28) Complex Obligation (BEST)

(29) Obligation:

TRS:
Rules:
U101(tt, M, N) → U102(isNatKind(activate(M)), activate(M), activate(N))
U102(tt, M, N) → U103(isNat(activate(N)), activate(M), activate(N))
U103(tt, M, N) → U104(isNatKind(activate(N)), activate(M), activate(N))
U104(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V1, V2) → U32(isNatKind(activate(V1)), activate(V1), activate(V2))
U32(tt, V1, V2) → U33(isNatKind(activate(V2)), activate(V1), activate(V2))
U33(tt, V1, V2) → U34(isNatKind(activate(V2)), activate(V1), activate(V2))
U34(tt, V1, V2) → U35(isNat(activate(V1)), activate(V2))
U35(tt, V2) → U36(isNat(activate(V2)))
U36(tt) → tt
U41(tt, V2) → U42(isNatKind(activate(V2)))
U42(tt) → tt
U51(tt) → tt
U61(tt, V2) → U62(isNatKind(activate(V2)))
U62(tt) → tt
U71(tt, N) → U72(isNatKind(activate(N)), activate(N))
U72(tt, N) → activate(N)
U81(tt, M, N) → U82(isNatKind(activate(M)), activate(M), activate(N))
U82(tt, M, N) → U83(isNat(activate(N)), activate(M), activate(N))
U83(tt, M, N) → U84(isNatKind(activate(N)), activate(M), activate(N))
U84(tt, M, N) → s(plus(activate(N), activate(M)))
U91(tt, N) → U92(isNatKind(activate(N)))
U92(tt) → 0'
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__x(V1, V2)) → U31(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U41(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U51(isNatKind(activate(V1)))
isNatKind(n__x(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
plus(N, 0') → U71(isNat(N), N)
plus(N, s(M)) → U81(isNat(M), M, N)
x(N, 0') → U91(isNat(N), N)
x(N, s(M)) → U101(isNat(M), M, N)
0' → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0'
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(X2))
activate(X) → X

Types:
U101 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
tt :: tt
U102 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
isNatKind :: n__0:n__plus:n__s:n__x → tt
activate :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U103 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
isNat :: n__0:n__plus:n__s:n__x → tt
U104 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U11 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U12 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U13 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U14 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U15 :: tt → n__0:n__plus:n__s:n__x → tt
U16 :: tt → tt
U21 :: tt → n__0:n__plus:n__s:n__x → tt
U22 :: tt → n__0:n__plus:n__s:n__x → tt
U23 :: tt → tt
U31 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U32 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U33 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U34 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U35 :: tt → n__0:n__plus:n__s:n__x → tt
U36 :: tt → tt
U41 :: tt → n__0:n__plus:n__s:n__x → tt
U42 :: tt → tt
U51 :: tt → tt
U61 :: tt → n__0:n__plus:n__s:n__x → tt
U62 :: tt → tt
U71 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U72 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U81 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U82 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U83 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U84 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U91 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U92 :: tt → n__0:n__plus:n__s:n__x
0' :: n__0:n__plus:n__s:n__x
n__0 :: n__0:n__plus:n__s:n__x
n__plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
hole_n__0:n__plus:n__s:n__x1_7 :: n__0:n__plus:n__s:n__x
hole_tt2_7 :: tt
gen_n__0:n__plus:n__s:n__x3_7 :: Nat → n__0:n__plus:n__s:n__x

Lemmas:
activate(gen_n__0:n__plus:n__s:n__x3_7(n18510_7)) → gen_n__0:n__plus:n__s:n__x3_7(n18510_7), rt ∈ Ω(1 + n18510₇)
isNatKind(gen_n__0:n__plus:n__s:n__x3_7(n14928_7)) → tt, rt ∈ Ω(1 + n14928₇ + n14928₇²)

Generator Equations:
gen_n__0:n__plus:n__s:n__x3_7(0) ⇔ n__0
gen_n__0:n__plus:n__s:n__x3_7(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s:n__x3_7(x), n__0)

The following defined symbols remain to be analysed:
plus, U71, U72

They will be analysed ascendingly in the following order:
isNatKind = activate
isNatKind = isNat
isNatKind = plus
isNatKind = x
isNatKind = U71
isNatKind = U72
isNatKind = U91
activate = isNat
activate = plus
activate = x
activate = U71
activate = U72
activate = U91
isNat = plus
isNat = x
isNat = U71
isNat = U72
isNat = U91
plus = x
plus = U71
plus = U72
plus = U91
x = U71
x = U72
x = U91
U71 = U72
U71 = U91
U72 = U91

(30) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(31) Obligation:

TRS:
Rules:
U101(tt, M, N) → U102(isNatKind(activate(M)), activate(M), activate(N))
U102(tt, M, N) → U103(isNat(activate(N)), activate(M), activate(N))
U103(tt, M, N) → U104(isNatKind(activate(N)), activate(M), activate(N))
U104(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V1, V2) → U32(isNatKind(activate(V1)), activate(V1), activate(V2))
U32(tt, V1, V2) → U33(isNatKind(activate(V2)), activate(V1), activate(V2))
U33(tt, V1, V2) → U34(isNatKind(activate(V2)), activate(V1), activate(V2))
U34(tt, V1, V2) → U35(isNat(activate(V1)), activate(V2))
U35(tt, V2) → U36(isNat(activate(V2)))
U36(tt) → tt
U41(tt, V2) → U42(isNatKind(activate(V2)))
U42(tt) → tt
U51(tt) → tt
U61(tt, V2) → U62(isNatKind(activate(V2)))
U62(tt) → tt
U71(tt, N) → U72(isNatKind(activate(N)), activate(N))
U72(tt, N) → activate(N)
U81(tt, M, N) → U82(isNatKind(activate(M)), activate(M), activate(N))
U82(tt, M, N) → U83(isNat(activate(N)), activate(M), activate(N))
U83(tt, M, N) → U84(isNatKind(activate(N)), activate(M), activate(N))
U84(tt, M, N) → s(plus(activate(N), activate(M)))
U91(tt, N) → U92(isNatKind(activate(N)))
U92(tt) → 0'
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__x(V1, V2)) → U31(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U41(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U51(isNatKind(activate(V1)))
isNatKind(n__x(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
plus(N, 0') → U71(isNat(N), N)
plus(N, s(M)) → U81(isNat(M), M, N)
x(N, 0') → U91(isNat(N), N)
x(N, s(M)) → U101(isNat(M), M, N)
0' → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0'
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(X2))
activate(X) → X

Types:
U101 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
tt :: tt
U102 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
isNatKind :: n__0:n__plus:n__s:n__x → tt
activate :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U103 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
isNat :: n__0:n__plus:n__s:n__x → tt
U104 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U11 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U12 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U13 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U14 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U15 :: tt → n__0:n__plus:n__s:n__x → tt
U16 :: tt → tt
U21 :: tt → n__0:n__plus:n__s:n__x → tt
U22 :: tt → n__0:n__plus:n__s:n__x → tt
U23 :: tt → tt
U31 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U32 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U33 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U34 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U35 :: tt → n__0:n__plus:n__s:n__x → tt
U36 :: tt → tt
U41 :: tt → n__0:n__plus:n__s:n__x → tt
U42 :: tt → tt
U51 :: tt → tt
U61 :: tt → n__0:n__plus:n__s:n__x → tt
U62 :: tt → tt
U71 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U72 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U81 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U82 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U83 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U84 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U91 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U92 :: tt → n__0:n__plus:n__s:n__x
0' :: n__0:n__plus:n__s:n__x
n__0 :: n__0:n__plus:n__s:n__x
n__plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
hole_n__0:n__plus:n__s:n__x1_7 :: n__0:n__plus:n__s:n__x
hole_tt2_7 :: tt
gen_n__0:n__plus:n__s:n__x3_7 :: Nat → n__0:n__plus:n__s:n__x

Lemmas:
activate(gen_n__0:n__plus:n__s:n__x3_7(n18510_7)) → gen_n__0:n__plus:n__s:n__x3_7(n18510_7), rt ∈ Ω(1 + n18510₇)
isNatKind(gen_n__0:n__plus:n__s:n__x3_7(n14928_7)) → tt, rt ∈ Ω(1 + n14928₇ + n14928₇²)

Generator Equations:
gen_n__0:n__plus:n__s:n__x3_7(0) ⇔ n__0
gen_n__0:n__plus:n__s:n__x3_7(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s:n__x3_7(x), n__0)

The following defined symbols remain to be analysed:
U71, U72

They will be analysed ascendingly in the following order:
isNatKind = activate
isNatKind = isNat
isNatKind = plus
isNatKind = x
isNatKind = U71
isNatKind = U72
isNatKind = U91
activate = isNat
activate = plus
activate = x
activate = U71
activate = U72
activate = U91
isNat = plus
isNat = x
isNat = U71
isNat = U72
isNat = U91
plus = x
plus = U71
plus = U72
plus = U91
x = U71
x = U72
x = U91
U71 = U72
U71 = U91
U72 = U91

(32) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U71.

(33) Obligation:

TRS:
Rules:
U101(tt, M, N) → U102(isNatKind(activate(M)), activate(M), activate(N))
U102(tt, M, N) → U103(isNat(activate(N)), activate(M), activate(N))
U103(tt, M, N) → U104(isNatKind(activate(N)), activate(M), activate(N))
U104(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V1, V2) → U32(isNatKind(activate(V1)), activate(V1), activate(V2))
U32(tt, V1, V2) → U33(isNatKind(activate(V2)), activate(V1), activate(V2))
U33(tt, V1, V2) → U34(isNatKind(activate(V2)), activate(V1), activate(V2))
U34(tt, V1, V2) → U35(isNat(activate(V1)), activate(V2))
U35(tt, V2) → U36(isNat(activate(V2)))
U36(tt) → tt
U41(tt, V2) → U42(isNatKind(activate(V2)))
U42(tt) → tt
U51(tt) → tt
U61(tt, V2) → U62(isNatKind(activate(V2)))
U62(tt) → tt
U71(tt, N) → U72(isNatKind(activate(N)), activate(N))
U72(tt, N) → activate(N)
U81(tt, M, N) → U82(isNatKind(activate(M)), activate(M), activate(N))
U82(tt, M, N) → U83(isNat(activate(N)), activate(M), activate(N))
U83(tt, M, N) → U84(isNatKind(activate(N)), activate(M), activate(N))
U84(tt, M, N) → s(plus(activate(N), activate(M)))
U91(tt, N) → U92(isNatKind(activate(N)))
U92(tt) → 0'
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__x(V1, V2)) → U31(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U41(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U51(isNatKind(activate(V1)))
isNatKind(n__x(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
plus(N, 0') → U71(isNat(N), N)
plus(N, s(M)) → U81(isNat(M), M, N)
x(N, 0') → U91(isNat(N), N)
x(N, s(M)) → U101(isNat(M), M, N)
0' → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0'
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(X2))
activate(X) → X

Types:
U101 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
tt :: tt
U102 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
isNatKind :: n__0:n__plus:n__s:n__x → tt
activate :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U103 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
isNat :: n__0:n__plus:n__s:n__x → tt
U104 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U11 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U12 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U13 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U14 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U15 :: tt → n__0:n__plus:n__s:n__x → tt
U16 :: tt → tt
U21 :: tt → n__0:n__plus:n__s:n__x → tt
U22 :: tt → n__0:n__plus:n__s:n__x → tt
U23 :: tt → tt
U31 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U32 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U33 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U34 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U35 :: tt → n__0:n__plus:n__s:n__x → tt
U36 :: tt → tt
U41 :: tt → n__0:n__plus:n__s:n__x → tt
U42 :: tt → tt
U51 :: tt → tt
U61 :: tt → n__0:n__plus:n__s:n__x → tt
U62 :: tt → tt
U71 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U72 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U81 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U82 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U83 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U84 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U91 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U92 :: tt → n__0:n__plus:n__s:n__x
0' :: n__0:n__plus:n__s:n__x
n__0 :: n__0:n__plus:n__s:n__x
n__plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
hole_n__0:n__plus:n__s:n__x1_7 :: n__0:n__plus:n__s:n__x
hole_tt2_7 :: tt
gen_n__0:n__plus:n__s:n__x3_7 :: Nat → n__0:n__plus:n__s:n__x

Lemmas:
activate(gen_n__0:n__plus:n__s:n__x3_7(n18510_7)) → gen_n__0:n__plus:n__s:n__x3_7(n18510_7), rt ∈ Ω(1 + n18510₇)
isNatKind(gen_n__0:n__plus:n__s:n__x3_7(n14928_7)) → tt, rt ∈ Ω(1 + n14928₇ + n14928₇²)

Generator Equations:
gen_n__0:n__plus:n__s:n__x3_7(0) ⇔ n__0
gen_n__0:n__plus:n__s:n__x3_7(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s:n__x3_7(x), n__0)

The following defined symbols remain to be analysed:
U72

They will be analysed ascendingly in the following order:
isNatKind = activate
isNatKind = isNat
isNatKind = plus
isNatKind = x
isNatKind = U71
isNatKind = U72
isNatKind = U91
activate = isNat
activate = plus
activate = x
activate = U71
activate = U72
activate = U91
isNat = plus
isNat = x
isNat = U71
isNat = U72
isNat = U91
plus = x
plus = U71
plus = U72
plus = U91
x = U71
x = U72
x = U91
U71 = U72
U71 = U91
U72 = U91

(34) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U72.

(35) Obligation:

TRS:
Rules:
U101(tt, M, N) → U102(isNatKind(activate(M)), activate(M), activate(N))
U102(tt, M, N) → U103(isNat(activate(N)), activate(M), activate(N))
U103(tt, M, N) → U104(isNatKind(activate(N)), activate(M), activate(N))
U104(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V1, V2) → U32(isNatKind(activate(V1)), activate(V1), activate(V2))
U32(tt, V1, V2) → U33(isNatKind(activate(V2)), activate(V1), activate(V2))
U33(tt, V1, V2) → U34(isNatKind(activate(V2)), activate(V1), activate(V2))
U34(tt, V1, V2) → U35(isNat(activate(V1)), activate(V2))
U35(tt, V2) → U36(isNat(activate(V2)))
U36(tt) → tt
U41(tt, V2) → U42(isNatKind(activate(V2)))
U42(tt) → tt
U51(tt) → tt
U61(tt, V2) → U62(isNatKind(activate(V2)))
U62(tt) → tt
U71(tt, N) → U72(isNatKind(activate(N)), activate(N))
U72(tt, N) → activate(N)
U81(tt, M, N) → U82(isNatKind(activate(M)), activate(M), activate(N))
U82(tt, M, N) → U83(isNat(activate(N)), activate(M), activate(N))
U83(tt, M, N) → U84(isNatKind(activate(N)), activate(M), activate(N))
U84(tt, M, N) → s(plus(activate(N), activate(M)))
U91(tt, N) → U92(isNatKind(activate(N)))
U92(tt) → 0'
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__x(V1, V2)) → U31(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U41(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U51(isNatKind(activate(V1)))
isNatKind(n__x(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
plus(N, 0') → U71(isNat(N), N)
plus(N, s(M)) → U81(isNat(M), M, N)
x(N, 0') → U91(isNat(N), N)
x(N, s(M)) → U101(isNat(M), M, N)
0' → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0'
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(X2))
activate(X) → X

Types:
U101 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
tt :: tt
U102 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
isNatKind :: n__0:n__plus:n__s:n__x → tt
activate :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U103 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
isNat :: n__0:n__plus:n__s:n__x → tt
U104 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U11 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U12 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U13 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U14 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U15 :: tt → n__0:n__plus:n__s:n__x → tt
U16 :: tt → tt
U21 :: tt → n__0:n__plus:n__s:n__x → tt
U22 :: tt → n__0:n__plus:n__s:n__x → tt
U23 :: tt → tt
U31 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U32 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U33 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U34 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U35 :: tt → n__0:n__plus:n__s:n__x → tt
U36 :: tt → tt
U41 :: tt → n__0:n__plus:n__s:n__x → tt
U42 :: tt → tt
U51 :: tt → tt
U61 :: tt → n__0:n__plus:n__s:n__x → tt
U62 :: tt → tt
U71 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U72 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U81 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U82 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U83 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U84 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U91 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U92 :: tt → n__0:n__plus:n__s:n__x
0' :: n__0:n__plus:n__s:n__x
n__0 :: n__0:n__plus:n__s:n__x
n__plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
hole_n__0:n__plus:n__s:n__x1_7 :: n__0:n__plus:n__s:n__x
hole_tt2_7 :: tt
gen_n__0:n__plus:n__s:n__x3_7 :: Nat → n__0:n__plus:n__s:n__x

Lemmas:
activate(gen_n__0:n__plus:n__s:n__x3_7(n18510_7)) → gen_n__0:n__plus:n__s:n__x3_7(n18510_7), rt ∈ Ω(1 + n18510₇)
isNatKind(gen_n__0:n__plus:n__s:n__x3_7(n14928_7)) → tt, rt ∈ Ω(1 + n14928₇ + n14928₇²)

Generator Equations:
gen_n__0:n__plus:n__s:n__x3_7(0) ⇔ n__0
gen_n__0:n__plus:n__s:n__x3_7(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s:n__x3_7(x), n__0)

No more defined symbols left to analyse.

(36) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
isNatKind(gen_n__0:n__plus:n__s:n__x3_7(n14928_7)) → tt, rt ∈ Ω(1 + n14928₇ + n14928₇²)

(37) BOUNDS(n^2, INF)

(38) Obligation:

TRS:
Rules:
U101(tt, M, N) → U102(isNatKind(activate(M)), activate(M), activate(N))
U102(tt, M, N) → U103(isNat(activate(N)), activate(M), activate(N))
U103(tt, M, N) → U104(isNatKind(activate(N)), activate(M), activate(N))
U104(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V1, V2) → U32(isNatKind(activate(V1)), activate(V1), activate(V2))
U32(tt, V1, V2) → U33(isNatKind(activate(V2)), activate(V1), activate(V2))
U33(tt, V1, V2) → U34(isNatKind(activate(V2)), activate(V1), activate(V2))
U34(tt, V1, V2) → U35(isNat(activate(V1)), activate(V2))
U35(tt, V2) → U36(isNat(activate(V2)))
U36(tt) → tt
U41(tt, V2) → U42(isNatKind(activate(V2)))
U42(tt) → tt
U51(tt) → tt
U61(tt, V2) → U62(isNatKind(activate(V2)))
U62(tt) → tt
U71(tt, N) → U72(isNatKind(activate(N)), activate(N))
U72(tt, N) → activate(N)
U81(tt, M, N) → U82(isNatKind(activate(M)), activate(M), activate(N))
U82(tt, M, N) → U83(isNat(activate(N)), activate(M), activate(N))
U83(tt, M, N) → U84(isNatKind(activate(N)), activate(M), activate(N))
U84(tt, M, N) → s(plus(activate(N), activate(M)))
U91(tt, N) → U92(isNatKind(activate(N)))
U92(tt) → 0'
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__x(V1, V2)) → U31(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U41(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U51(isNatKind(activate(V1)))
isNatKind(n__x(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
plus(N, 0') → U71(isNat(N), N)
plus(N, s(M)) → U81(isNat(M), M, N)
x(N, 0') → U91(isNat(N), N)
x(N, s(M)) → U101(isNat(M), M, N)
0' → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0'
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(X2))
activate(X) → X

Types:
U101 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
tt :: tt
U102 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
isNatKind :: n__0:n__plus:n__s:n__x → tt
activate :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U103 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
isNat :: n__0:n__plus:n__s:n__x → tt
U104 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U11 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U12 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U13 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U14 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U15 :: tt → n__0:n__plus:n__s:n__x → tt
U16 :: tt → tt
U21 :: tt → n__0:n__plus:n__s:n__x → tt
U22 :: tt → n__0:n__plus:n__s:n__x → tt
U23 :: tt → tt
U31 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U32 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U33 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U34 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U35 :: tt → n__0:n__plus:n__s:n__x → tt
U36 :: tt → tt
U41 :: tt → n__0:n__plus:n__s:n__x → tt
U42 :: tt → tt
U51 :: tt → tt
U61 :: tt → n__0:n__plus:n__s:n__x → tt
U62 :: tt → tt
U71 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U72 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U81 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U82 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U83 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U84 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U91 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U92 :: tt → n__0:n__plus:n__s:n__x
0' :: n__0:n__plus:n__s:n__x
n__0 :: n__0:n__plus:n__s:n__x
n__plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
hole_n__0:n__plus:n__s:n__x1_7 :: n__0:n__plus:n__s:n__x
hole_tt2_7 :: tt
gen_n__0:n__plus:n__s:n__x3_7 :: Nat → n__0:n__plus:n__s:n__x

Lemmas:
activate(gen_n__0:n__plus:n__s:n__x3_7(n18510_7)) → gen_n__0:n__plus:n__s:n__x3_7(n18510_7), rt ∈ Ω(1 + n18510₇)
isNatKind(gen_n__0:n__plus:n__s:n__x3_7(n14928_7)) → tt, rt ∈ Ω(1 + n14928₇ + n14928₇²)

Generator Equations:
gen_n__0:n__plus:n__s:n__x3_7(0) ⇔ n__0
gen_n__0:n__plus:n__s:n__x3_7(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s:n__x3_7(x), n__0)

No more defined symbols left to analyse.

(39) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
isNatKind(gen_n__0:n__plus:n__s:n__x3_7(n14928_7)) → tt, rt ∈ Ω(1 + n14928₇ + n14928₇²)

(40) BOUNDS(n^2, INF)

(41) Obligation:

TRS:
Rules:
U101(tt, M, N) → U102(isNatKind(activate(M)), activate(M), activate(N))
U102(tt, M, N) → U103(isNat(activate(N)), activate(M), activate(N))
U103(tt, M, N) → U104(isNatKind(activate(N)), activate(M), activate(N))
U104(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V1, V2) → U32(isNatKind(activate(V1)), activate(V1), activate(V2))
U32(tt, V1, V2) → U33(isNatKind(activate(V2)), activate(V1), activate(V2))
U33(tt, V1, V2) → U34(isNatKind(activate(V2)), activate(V1), activate(V2))
U34(tt, V1, V2) → U35(isNat(activate(V1)), activate(V2))
U35(tt, V2) → U36(isNat(activate(V2)))
U36(tt) → tt
U41(tt, V2) → U42(isNatKind(activate(V2)))
U42(tt) → tt
U51(tt) → tt
U61(tt, V2) → U62(isNatKind(activate(V2)))
U62(tt) → tt
U71(tt, N) → U72(isNatKind(activate(N)), activate(N))
U72(tt, N) → activate(N)
U81(tt, M, N) → U82(isNatKind(activate(M)), activate(M), activate(N))
U82(tt, M, N) → U83(isNat(activate(N)), activate(M), activate(N))
U83(tt, M, N) → U84(isNatKind(activate(N)), activate(M), activate(N))
U84(tt, M, N) → s(plus(activate(N), activate(M)))
U91(tt, N) → U92(isNatKind(activate(N)))
U92(tt) → 0'
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__x(V1, V2)) → U31(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U41(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U51(isNatKind(activate(V1)))
isNatKind(n__x(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
plus(N, 0') → U71(isNat(N), N)
plus(N, s(M)) → U81(isNat(M), M, N)
x(N, 0') → U91(isNat(N), N)
x(N, s(M)) → U101(isNat(M), M, N)
0' → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0'
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(X2))
activate(X) → X

Types:
U101 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
tt :: tt
U102 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
isNatKind :: n__0:n__plus:n__s:n__x → tt
activate :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U103 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
isNat :: n__0:n__plus:n__s:n__x → tt
U104 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U11 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U12 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U13 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U14 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U15 :: tt → n__0:n__plus:n__s:n__x → tt
U16 :: tt → tt
U21 :: tt → n__0:n__plus:n__s:n__x → tt
U22 :: tt → n__0:n__plus:n__s:n__x → tt
U23 :: tt → tt
U31 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U32 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U33 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U34 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U35 :: tt → n__0:n__plus:n__s:n__x → tt
U36 :: tt → tt
U41 :: tt → n__0:n__plus:n__s:n__x → tt
U42 :: tt → tt
U51 :: tt → tt
U61 :: tt → n__0:n__plus:n__s:n__x → tt
U62 :: tt → tt
U71 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U72 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U81 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U82 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U83 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U84 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U91 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U92 :: tt → n__0:n__plus:n__s:n__x
0' :: n__0:n__plus:n__s:n__x
n__0 :: n__0:n__plus:n__s:n__x
n__plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
hole_n__0:n__plus:n__s:n__x1_7 :: n__0:n__plus:n__s:n__x
hole_tt2_7 :: tt
gen_n__0:n__plus:n__s:n__x3_7 :: Nat → n__0:n__plus:n__s:n__x

Lemmas:
activate(gen_n__0:n__plus:n__s:n__x3_7(n5_7)) → gen_n__0:n__plus:n__s:n__x3_7(n5_7), rt ∈ Ω(1 + n5₇)
isNatKind(gen_n__0:n__plus:n__s:n__x3_7(n14928_7)) → tt, rt ∈ Ω(1 + n14928₇ + n14928₇²)

Generator Equations:
gen_n__0:n__plus:n__s:n__x3_7(0) ⇔ n__0
gen_n__0:n__plus:n__s:n__x3_7(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s:n__x3_7(x), n__0)

No more defined symbols left to analyse.

(42) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
isNatKind(gen_n__0:n__plus:n__s:n__x3_7(n14928_7)) → tt, rt ∈ Ω(1 + n14928₇ + n14928₇²)

(43) BOUNDS(n^2, INF)

(44) Obligation:

TRS:
Rules:
U101(tt, M, N) → U102(isNatKind(activate(M)), activate(M), activate(N))
U102(tt, M, N) → U103(isNat(activate(N)), activate(M), activate(N))
U103(tt, M, N) → U104(isNatKind(activate(N)), activate(M), activate(N))
U104(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V1, V2) → U32(isNatKind(activate(V1)), activate(V1), activate(V2))
U32(tt, V1, V2) → U33(isNatKind(activate(V2)), activate(V1), activate(V2))
U33(tt, V1, V2) → U34(isNatKind(activate(V2)), activate(V1), activate(V2))
U34(tt, V1, V2) → U35(isNat(activate(V1)), activate(V2))
U35(tt, V2) → U36(isNat(activate(V2)))
U36(tt) → tt
U41(tt, V2) → U42(isNatKind(activate(V2)))
U42(tt) → tt
U51(tt) → tt
U61(tt, V2) → U62(isNatKind(activate(V2)))
U62(tt) → tt
U71(tt, N) → U72(isNatKind(activate(N)), activate(N))
U72(tt, N) → activate(N)
U81(tt, M, N) → U82(isNatKind(activate(M)), activate(M), activate(N))
U82(tt, M, N) → U83(isNat(activate(N)), activate(M), activate(N))
U83(tt, M, N) → U84(isNatKind(activate(N)), activate(M), activate(N))
U84(tt, M, N) → s(plus(activate(N), activate(M)))
U91(tt, N) → U92(isNatKind(activate(N)))
U92(tt) → 0'
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNat(n__x(V1, V2)) → U31(isNatKind(activate(V1)), activate(V1), activate(V2))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U41(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U51(isNatKind(activate(V1)))
isNatKind(n__x(V1, V2)) → U61(isNatKind(activate(V1)), activate(V2))
plus(N, 0') → U71(isNat(N), N)
plus(N, s(M)) → U81(isNat(M), M, N)
x(N, 0') → U91(isNat(N), N)
x(N, s(M)) → U101(isNat(M), M, N)
0' → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0'
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(X2))
activate(X) → X

Types:
U101 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
tt :: tt
U102 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
isNatKind :: n__0:n__plus:n__s:n__x → tt
activate :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U103 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
isNat :: n__0:n__plus:n__s:n__x → tt
U104 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U11 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U12 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U13 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U14 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U15 :: tt → n__0:n__plus:n__s:n__x → tt
U16 :: tt → tt
U21 :: tt → n__0:n__plus:n__s:n__x → tt
U22 :: tt → n__0:n__plus:n__s:n__x → tt
U23 :: tt → tt
U31 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U32 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U33 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U34 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → tt
U35 :: tt → n__0:n__plus:n__s:n__x → tt
U36 :: tt → tt
U41 :: tt → n__0:n__plus:n__s:n__x → tt
U42 :: tt → tt
U51 :: tt → tt
U61 :: tt → n__0:n__plus:n__s:n__x → tt
U62 :: tt → tt
U71 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U72 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U81 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U82 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U83 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U84 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U91 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U92 :: tt → n__0:n__plus:n__s:n__x
0' :: n__0:n__plus:n__s:n__x
n__0 :: n__0:n__plus:n__s:n__x
n__plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
hole_n__0:n__plus:n__s:n__x1_7 :: n__0:n__plus:n__s:n__x
hole_tt2_7 :: tt
gen_n__0:n__plus:n__s:n__x3_7 :: Nat → n__0:n__plus:n__s:n__x

Lemmas:
activate(gen_n__0:n__plus:n__s:n__x3_7(n5_7)) → gen_n__0:n__plus:n__s:n__x3_7(n5_7), rt ∈ Ω(1 + n5₇)

Generator Equations:
gen_n__0:n__plus:n__s:n__x3_7(0) ⇔ n__0
gen_n__0:n__plus:n__s:n__x3_7(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s:n__x3_7(x), n__0)

No more defined symbols left to analyse.

(45) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
activate(gen_n__0:n__plus:n__s:n__x3_7(n5_7)) → gen_n__0:n__plus:n__s:n__x3_7(n5_7), rt ∈ Ω(1 + n5₇)

(46) BOUNDS(n^1, INF)