The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(1)).

(0) Obligation:

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

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

Rewrite Strategy: INNERMOST

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0, z1)) → mark(U32(isNat(z0), z1))
active(U32(tt, z0)) → mark(U33(isNat(z0)))
active(U33(tt)) → mark(tt)
active(U41(tt, z0)) → mark(z0)
active(U51(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U61(tt)) → mark(0)
active(U71(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNat(x(z0, z1))) → mark(U31(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(isNatKind(x(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(plus(z0, 0)) → mark(U41(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U51(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(x(z0, 0)) → mark(U61(and(isNat(z0), isNatKind(z0))))
active(x(z0, s(z1))) → mark(U71(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
mark(U11(z0, z1, z2)) → active(U11(mark(z0), z1, z2))
mark(tt) → active(tt)
mark(U12(z0, z1)) → active(U12(mark(z0), z1))
mark(isNat(z0)) → active(isNat(z0))
mark(U13(z0)) → active(U13(mark(z0)))
mark(U21(z0, z1)) → active(U21(mark(z0), z1))
mark(U22(z0)) → active(U22(mark(z0)))
mark(U31(z0, z1, z2)) → active(U31(mark(z0), z1, z2))
mark(U32(z0, z1)) → active(U32(mark(z0), z1))
mark(U33(z0)) → active(U33(mark(z0)))
mark(U41(z0, z1)) → active(U41(mark(z0), z1))
mark(U51(z0, z1, z2)) → active(U51(mark(z0), z1, z2))
mark(s(z0)) → active(s(mark(z0)))
mark(plus(z0, z1)) → active(plus(mark(z0), mark(z1)))
mark(U61(z0)) → active(U61(mark(z0)))
mark(0) → active(0)
mark(U71(z0, z1, z2)) → active(U71(mark(z0), z1, z2))
mark(x(z0, z1)) → active(x(mark(z0), mark(z1)))
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(isNatKind(z0)) → active(isNatKind(z0))
U11(mark(z0), z1, z2) → U11(z0, z1, z2)
U11(z0, mark(z1), z2) → U11(z0, z1, z2)
U11(z0, z1, mark(z2)) → U11(z0, z1, z2)
U11(active(z0), z1, z2) → U11(z0, z1, z2)
U11(z0, active(z1), z2) → U11(z0, z1, z2)
U11(z0, z1, active(z2)) → U11(z0, z1, z2)
U12(mark(z0), z1) → U12(z0, z1)
U12(z0, mark(z1)) → U12(z0, z1)
U12(active(z0), z1) → U12(z0, z1)
U12(z0, active(z1)) → U12(z0, z1)
isNat(mark(z0)) → isNat(z0)
isNat(active(z0)) → isNat(z0)
U13(mark(z0)) → U13(z0)
U13(active(z0)) → U13(z0)
U21(mark(z0), z1) → U21(z0, z1)
U21(z0, mark(z1)) → U21(z0, z1)
U21(active(z0), z1) → U21(z0, z1)
U21(z0, active(z1)) → U21(z0, z1)
U22(mark(z0)) → U22(z0)
U22(active(z0)) → U22(z0)
U31(mark(z0), z1, z2) → U31(z0, z1, z2)
U31(z0, mark(z1), z2) → U31(z0, z1, z2)
U31(z0, z1, mark(z2)) → U31(z0, z1, z2)
U31(active(z0), z1, z2) → U31(z0, z1, z2)
U31(z0, active(z1), z2) → U31(z0, z1, z2)
U31(z0, z1, active(z2)) → U31(z0, z1, z2)
U32(mark(z0), z1) → U32(z0, z1)
U32(z0, mark(z1)) → U32(z0, z1)
U32(active(z0), z1) → U32(z0, z1)
U32(z0, active(z1)) → U32(z0, z1)
U33(mark(z0)) → U33(z0)
U33(active(z0)) → U33(z0)
U41(mark(z0), z1) → U41(z0, z1)
U41(z0, mark(z1)) → U41(z0, z1)
U41(active(z0), z1) → U41(z0, z1)
U41(z0, active(z1)) → U41(z0, z1)
U51(mark(z0), z1, z2) → U51(z0, z1, z2)
U51(z0, mark(z1), z2) → U51(z0, z1, z2)
U51(z0, z1, mark(z2)) → U51(z0, z1, z2)
U51(active(z0), z1, z2) → U51(z0, z1, z2)
U51(z0, active(z1), z2) → U51(z0, z1, z2)
U51(z0, z1, active(z2)) → U51(z0, z1, z2)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
plus(mark(z0), z1) → plus(z0, z1)
plus(z0, mark(z1)) → plus(z0, z1)
plus(active(z0), z1) → plus(z0, z1)
plus(z0, active(z1)) → plus(z0, z1)
U61(mark(z0)) → U61(z0)
U61(active(z0)) → U61(z0)
U71(mark(z0), z1, z2) → U71(z0, z1, z2)
U71(z0, mark(z1), z2) → U71(z0, z1, z2)
U71(z0, z1, mark(z2)) → U71(z0, z1, z2)
U71(active(z0), z1, z2) → U71(z0, z1, z2)
U71(z0, active(z1), z2) → U71(z0, z1, z2)
U71(z0, z1, active(z2)) → U71(z0, z1, z2)
x(mark(z0), z1) → x(z0, z1)
x(z0, mark(z1)) → x(z0, z1)
x(active(z0), z1) → x(z0, z1)
x(z0, active(z1)) → x(z0, z1)
and(mark(z0), z1) → and(z0, z1)
and(z0, mark(z1)) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
isNatKind(mark(z0)) → isNatKind(z0)
isNatKind(active(z0)) → isNatKind(z0)

Tuples:

ACTIVE(U11(tt, z0, z1)) → c(MARK(U12(isNat(z0), z1)), U12'(isNat(z0), z1), ISNAT(z0))
ACTIVE(U12(tt, z0)) → c1(MARK(U13(isNat(z0))), U13'(isNat(z0)), ISNAT(z0))
ACTIVE(U13(tt)) → c2(MARK(tt))
ACTIVE(U21(tt, z0)) → c3(MARK(U22(isNat(z0))), U22'(isNat(z0)), ISNAT(z0))
ACTIVE(U22(tt)) → c4(MARK(tt))
ACTIVE(U31(tt, z0, z1)) → c5(MARK(U32(isNat(z0), z1)), U32'(isNat(z0), z1), ISNAT(z0))
ACTIVE(U32(tt, z0)) → c6(MARK(U33(isNat(z0))), U33'(isNat(z0)), ISNAT(z0))
ACTIVE(U33(tt)) → c7(MARK(tt))
ACTIVE(U41(tt, z0)) → c8(MARK(z0))
ACTIVE(U51(tt, z0, z1)) → c9(MARK(s(plus(z1, z0))), S(plus(z1, z0)), PLUS(z1, z0))
ACTIVE(U61(tt)) → c10(MARK(0))
ACTIVE(U71(tt, z0, z1)) → c11(MARK(plus(x(z1, z0), z1)), PLUS(x(z1, z0), z1), X(z1, z0))
ACTIVE(and(tt, z0)) → c12(MARK(z0))
ACTIVE(isNat(0)) → c13(MARK(tt))
ACTIVE(isNat(plus(z0, z1))) → c14(MARK(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1), AND(isNatKind(z0), isNatKind(z1)), ISNATKIND(z0), ISNATKIND(z1))
ACTIVE(isNat(s(z0))) → c15(MARK(U21(isNatKind(z0), z0)), U21'(isNatKind(z0), z0), ISNATKIND(z0))
ACTIVE(isNat(x(z0, z1))) → c16(MARK(U31(and(isNatKind(z0), isNatKind(z1)), z0, z1)), U31'(and(isNatKind(z0), isNatKind(z1)), z0, z1), AND(isNatKind(z0), isNatKind(z1)), ISNATKIND(z0), ISNATKIND(z1))
ACTIVE(isNatKind(0)) → c17(MARK(tt))
ACTIVE(isNatKind(plus(z0, z1))) → c18(MARK(and(isNatKind(z0), isNatKind(z1))), AND(isNatKind(z0), isNatKind(z1)), ISNATKIND(z0), ISNATKIND(z1))
ACTIVE(isNatKind(s(z0))) → c19(MARK(isNatKind(z0)), ISNATKIND(z0))
ACTIVE(isNatKind(x(z0, z1))) → c20(MARK(and(isNatKind(z0), isNatKind(z1))), AND(isNatKind(z0), isNatKind(z1)), ISNATKIND(z0), ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c21(MARK(U41(and(isNat(z0), isNatKind(z0)), z0)), U41'(and(isNat(z0), isNatKind(z0)), z0), AND(isNat(z0), isNatKind(z0)), ISNAT(z0), ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c22(MARK(U51(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), U51'(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0), AND(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), AND(isNat(z1), isNatKind(z1)), ISNAT(z1), ISNATKIND(z1), AND(isNat(z0), isNatKind(z0)), ISNAT(z0), ISNATKIND(z0))
ACTIVE(x(z0, 0)) → c23(MARK(U61(and(isNat(z0), isNatKind(z0)))), U61'(and(isNat(z0), isNatKind(z0))), AND(isNat(z0), isNatKind(z0)), ISNAT(z0), ISNATKIND(z0))
ACTIVE(x(z0, s(z1))) → c24(MARK(U71(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), U71'(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0), AND(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), AND(isNat(z1), isNatKind(z1)), ISNAT(z1), ISNATKIND(z1), AND(isNat(z0), isNatKind(z0)), ISNAT(z0), ISNATKIND(z0))
MARK(U11(z0, z1, z2)) → c25(ACTIVE(U11(mark(z0), z1, z2)), U11'(mark(z0), z1, z2), MARK(z0))
MARK(tt) → c26(ACTIVE(tt))
MARK(U12(z0, z1)) → c27(ACTIVE(U12(mark(z0), z1)), U12'(mark(z0), z1), MARK(z0))
MARK(isNat(z0)) → c28(ACTIVE(isNat(z0)), ISNAT(z0))
MARK(U13(z0)) → c29(ACTIVE(U13(mark(z0))), U13'(mark(z0)), MARK(z0))
MARK(U21(z0, z1)) → c30(ACTIVE(U21(mark(z0), z1)), U21'(mark(z0), z1), MARK(z0))
MARK(U22(z0)) → c31(ACTIVE(U22(mark(z0))), U22'(mark(z0)), MARK(z0))
MARK(U31(z0, z1, z2)) → c32(ACTIVE(U31(mark(z0), z1, z2)), U31'(mark(z0), z1, z2), MARK(z0))
MARK(U32(z0, z1)) → c33(ACTIVE(U32(mark(z0), z1)), U32'(mark(z0), z1), MARK(z0))
MARK(U33(z0)) → c34(ACTIVE(U33(mark(z0))), U33'(mark(z0)), MARK(z0))
MARK(U41(z0, z1)) → c35(ACTIVE(U41(mark(z0), z1)), U41'(mark(z0), z1), MARK(z0))
MARK(U51(z0, z1, z2)) → c36(ACTIVE(U51(mark(z0), z1, z2)), U51'(mark(z0), z1, z2), MARK(z0))
MARK(s(z0)) → c37(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(plus(z0, z1)) → c38(ACTIVE(plus(mark(z0), mark(z1))), PLUS(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(U61(z0)) → c39(ACTIVE(U61(mark(z0))), U61'(mark(z0)), MARK(z0))
MARK(0) → c40(ACTIVE(0))
MARK(U71(z0, z1, z2)) → c41(ACTIVE(U71(mark(z0), z1, z2)), U71'(mark(z0), z1, z2), MARK(z0))
MARK(x(z0, z1)) → c42(ACTIVE(x(mark(z0), mark(z1))), X(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(and(z0, z1)) → c43(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(isNatKind(z0)) → c44(ACTIVE(isNatKind(z0)), ISNATKIND(z0))
U11'(mark(z0), z1, z2) → c45(U11'(z0, z1, z2))
U11'(z0, mark(z1), z2) → c46(U11'(z0, z1, z2))
U11'(z0, z1, mark(z2)) → c47(U11'(z0, z1, z2))
U11'(active(z0), z1, z2) → c48(U11'(z0, z1, z2))
U11'(z0, active(z1), z2) → c49(U11'(z0, z1, z2))
U11'(z0, z1, active(z2)) → c50(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c51(U12'(z0, z1))
U12'(z0, mark(z1)) → c52(U12'(z0, z1))
U12'(active(z0), z1) → c53(U12'(z0, z1))
U12'(z0, active(z1)) → c54(U12'(z0, z1))
ISNAT(mark(z0)) → c55(ISNAT(z0))
ISNAT(active(z0)) → c56(ISNAT(z0))
U13'(mark(z0)) → c57(U13'(z0))
U13'(active(z0)) → c58(U13'(z0))
U21'(mark(z0), z1) → c59(U21'(z0, z1))
U21'(z0, mark(z1)) → c60(U21'(z0, z1))
U21'(active(z0), z1) → c61(U21'(z0, z1))
U21'(z0, active(z1)) → c62(U21'(z0, z1))
U22'(mark(z0)) → c63(U22'(z0))
U22'(active(z0)) → c64(U22'(z0))
U31'(mark(z0), z1, z2) → c65(U31'(z0, z1, z2))
U31'(z0, mark(z1), z2) → c66(U31'(z0, z1, z2))
U31'(z0, z1, mark(z2)) → c67(U31'(z0, z1, z2))
U31'(active(z0), z1, z2) → c68(U31'(z0, z1, z2))
U31'(z0, active(z1), z2) → c69(U31'(z0, z1, z2))
U31'(z0, z1, active(z2)) → c70(U31'(z0, z1, z2))
U32'(mark(z0), z1) → c71(U32'(z0, z1))
U32'(z0, mark(z1)) → c72(U32'(z0, z1))
U32'(active(z0), z1) → c73(U32'(z0, z1))
U32'(z0, active(z1)) → c74(U32'(z0, z1))
U33'(mark(z0)) → c75(U33'(z0))
U33'(active(z0)) → c76(U33'(z0))
U41'(mark(z0), z1) → c77(U41'(z0, z1))
U41'(z0, mark(z1)) → c78(U41'(z0, z1))
U41'(active(z0), z1) → c79(U41'(z0, z1))
U41'(z0, active(z1)) → c80(U41'(z0, z1))
U51'(mark(z0), z1, z2) → c81(U51'(z0, z1, z2))
U51'(z0, mark(z1), z2) → c82(U51'(z0, z1, z2))
U51'(z0, z1, mark(z2)) → c83(U51'(z0, z1, z2))
U51'(active(z0), z1, z2) → c84(U51'(z0, z1, z2))
U51'(z0, active(z1), z2) → c85(U51'(z0, z1, z2))
U51'(z0, z1, active(z2)) → c86(U51'(z0, z1, z2))
S(mark(z0)) → c87(S(z0))
S(active(z0)) → c88(S(z0))
PLUS(mark(z0), z1) → c89(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c90(PLUS(z0, z1))
PLUS(active(z0), z1) → c91(PLUS(z0, z1))
PLUS(z0, active(z1)) → c92(PLUS(z0, z1))
U61'(mark(z0)) → c93(U61'(z0))
U61'(active(z0)) → c94(U61'(z0))
U71'(mark(z0), z1, z2) → c95(U71'(z0, z1, z2))
U71'(z0, mark(z1), z2) → c96(U71'(z0, z1, z2))
U71'(z0, z1, mark(z2)) → c97(U71'(z0, z1, z2))
U71'(active(z0), z1, z2) → c98(U71'(z0, z1, z2))
U71'(z0, active(z1), z2) → c99(U71'(z0, z1, z2))
U71'(z0, z1, active(z2)) → c100(U71'(z0, z1, z2))
X(mark(z0), z1) → c101(X(z0, z1))
X(z0, mark(z1)) → c102(X(z0, z1))
X(active(z0), z1) → c103(X(z0, z1))
X(z0, active(z1)) → c104(X(z0, z1))
AND(mark(z0), z1) → c105(AND(z0, z1))
AND(z0, mark(z1)) → c106(AND(z0, z1))
AND(active(z0), z1) → c107(AND(z0, z1))
AND(z0, active(z1)) → c108(AND(z0, z1))
ISNATKIND(mark(z0)) → c109(ISNATKIND(z0))
ISNATKIND(active(z0)) → c110(ISNATKIND(z0))

S tuples:

ACTIVE(U11(tt, z0, z1)) → c(MARK(U12(isNat(z0), z1)), U12'(isNat(z0), z1), ISNAT(z0))
ACTIVE(U12(tt, z0)) → c1(MARK(U13(isNat(z0))), U13'(isNat(z0)), ISNAT(z0))
ACTIVE(U13(tt)) → c2(MARK(tt))
ACTIVE(U21(tt, z0)) → c3(MARK(U22(isNat(z0))), U22'(isNat(z0)), ISNAT(z0))
ACTIVE(U22(tt)) → c4(MARK(tt))
ACTIVE(U31(tt, z0, z1)) → c5(MARK(U32(isNat(z0), z1)), U32'(isNat(z0), z1), ISNAT(z0))
ACTIVE(U32(tt, z0)) → c6(MARK(U33(isNat(z0))), U33'(isNat(z0)), ISNAT(z0))
ACTIVE(U33(tt)) → c7(MARK(tt))
ACTIVE(U41(tt, z0)) → c8(MARK(z0))
ACTIVE(U51(tt, z0, z1)) → c9(MARK(s(plus(z1, z0))), S(plus(z1, z0)), PLUS(z1, z0))
ACTIVE(U61(tt)) → c10(MARK(0))
ACTIVE(U71(tt, z0, z1)) → c11(MARK(plus(x(z1, z0), z1)), PLUS(x(z1, z0), z1), X(z1, z0))
ACTIVE(and(tt, z0)) → c12(MARK(z0))
ACTIVE(isNat(0)) → c13(MARK(tt))
ACTIVE(isNat(plus(z0, z1))) → c14(MARK(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1), AND(isNatKind(z0), isNatKind(z1)), ISNATKIND(z0), ISNATKIND(z1))
ACTIVE(isNat(s(z0))) → c15(MARK(U21(isNatKind(z0), z0)), U21'(isNatKind(z0), z0), ISNATKIND(z0))
ACTIVE(isNat(x(z0, z1))) → c16(MARK(U31(and(isNatKind(z0), isNatKind(z1)), z0, z1)), U31'(and(isNatKind(z0), isNatKind(z1)), z0, z1), AND(isNatKind(z0), isNatKind(z1)), ISNATKIND(z0), ISNATKIND(z1))
ACTIVE(isNatKind(0)) → c17(MARK(tt))
ACTIVE(isNatKind(plus(z0, z1))) → c18(MARK(and(isNatKind(z0), isNatKind(z1))), AND(isNatKind(z0), isNatKind(z1)), ISNATKIND(z0), ISNATKIND(z1))
ACTIVE(isNatKind(s(z0))) → c19(MARK(isNatKind(z0)), ISNATKIND(z0))
ACTIVE(isNatKind(x(z0, z1))) → c20(MARK(and(isNatKind(z0), isNatKind(z1))), AND(isNatKind(z0), isNatKind(z1)), ISNATKIND(z0), ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c21(MARK(U41(and(isNat(z0), isNatKind(z0)), z0)), U41'(and(isNat(z0), isNatKind(z0)), z0), AND(isNat(z0), isNatKind(z0)), ISNAT(z0), ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c22(MARK(U51(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), U51'(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0), AND(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), AND(isNat(z1), isNatKind(z1)), ISNAT(z1), ISNATKIND(z1), AND(isNat(z0), isNatKind(z0)), ISNAT(z0), ISNATKIND(z0))
ACTIVE(x(z0, 0)) → c23(MARK(U61(and(isNat(z0), isNatKind(z0)))), U61'(and(isNat(z0), isNatKind(z0))), AND(isNat(z0), isNatKind(z0)), ISNAT(z0), ISNATKIND(z0))
ACTIVE(x(z0, s(z1))) → c24(MARK(U71(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), U71'(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0), AND(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), AND(isNat(z1), isNatKind(z1)), ISNAT(z1), ISNATKIND(z1), AND(isNat(z0), isNatKind(z0)), ISNAT(z0), ISNATKIND(z0))
MARK(U11(z0, z1, z2)) → c25(ACTIVE(U11(mark(z0), z1, z2)), U11'(mark(z0), z1, z2), MARK(z0))
MARK(tt) → c26(ACTIVE(tt))
MARK(U12(z0, z1)) → c27(ACTIVE(U12(mark(z0), z1)), U12'(mark(z0), z1), MARK(z0))
MARK(isNat(z0)) → c28(ACTIVE(isNat(z0)), ISNAT(z0))
MARK(U13(z0)) → c29(ACTIVE(U13(mark(z0))), U13'(mark(z0)), MARK(z0))
MARK(U21(z0, z1)) → c30(ACTIVE(U21(mark(z0), z1)), U21'(mark(z0), z1), MARK(z0))
MARK(U22(z0)) → c31(ACTIVE(U22(mark(z0))), U22'(mark(z0)), MARK(z0))
MARK(U31(z0, z1, z2)) → c32(ACTIVE(U31(mark(z0), z1, z2)), U31'(mark(z0), z1, z2), MARK(z0))
MARK(U32(z0, z1)) → c33(ACTIVE(U32(mark(z0), z1)), U32'(mark(z0), z1), MARK(z0))
MARK(U33(z0)) → c34(ACTIVE(U33(mark(z0))), U33'(mark(z0)), MARK(z0))
MARK(U41(z0, z1)) → c35(ACTIVE(U41(mark(z0), z1)), U41'(mark(z0), z1), MARK(z0))
MARK(U51(z0, z1, z2)) → c36(ACTIVE(U51(mark(z0), z1, z2)), U51'(mark(z0), z1, z2), MARK(z0))
MARK(s(z0)) → c37(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(plus(z0, z1)) → c38(ACTIVE(plus(mark(z0), mark(z1))), PLUS(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(U61(z0)) → c39(ACTIVE(U61(mark(z0))), U61'(mark(z0)), MARK(z0))
MARK(0) → c40(ACTIVE(0))
MARK(U71(z0, z1, z2)) → c41(ACTIVE(U71(mark(z0), z1, z2)), U71'(mark(z0), z1, z2), MARK(z0))
MARK(x(z0, z1)) → c42(ACTIVE(x(mark(z0), mark(z1))), X(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(and(z0, z1)) → c43(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(isNatKind(z0)) → c44(ACTIVE(isNatKind(z0)), ISNATKIND(z0))
U11'(mark(z0), z1, z2) → c45(U11'(z0, z1, z2))
U11'(z0, mark(z1), z2) → c46(U11'(z0, z1, z2))
U11'(z0, z1, mark(z2)) → c47(U11'(z0, z1, z2))
U11'(active(z0), z1, z2) → c48(U11'(z0, z1, z2))
U11'(z0, active(z1), z2) → c49(U11'(z0, z1, z2))
U11'(z0, z1, active(z2)) → c50(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c51(U12'(z0, z1))
U12'(z0, mark(z1)) → c52(U12'(z0, z1))
U12'(active(z0), z1) → c53(U12'(z0, z1))
U12'(z0, active(z1)) → c54(U12'(z0, z1))
ISNAT(mark(z0)) → c55(ISNAT(z0))
ISNAT(active(z0)) → c56(ISNAT(z0))
U13'(mark(z0)) → c57(U13'(z0))
U13'(active(z0)) → c58(U13'(z0))
U21'(mark(z0), z1) → c59(U21'(z0, z1))
U21'(z0, mark(z1)) → c60(U21'(z0, z1))
U21'(active(z0), z1) → c61(U21'(z0, z1))
U21'(z0, active(z1)) → c62(U21'(z0, z1))
U22'(mark(z0)) → c63(U22'(z0))
U22'(active(z0)) → c64(U22'(z0))
U31'(mark(z0), z1, z2) → c65(U31'(z0, z1, z2))
U31'(z0, mark(z1), z2) → c66(U31'(z0, z1, z2))
U31'(z0, z1, mark(z2)) → c67(U31'(z0, z1, z2))
U31'(active(z0), z1, z2) → c68(U31'(z0, z1, z2))
U31'(z0, active(z1), z2) → c69(U31'(z0, z1, z2))
U31'(z0, z1, active(z2)) → c70(U31'(z0, z1, z2))
U32'(mark(z0), z1) → c71(U32'(z0, z1))
U32'(z0, mark(z1)) → c72(U32'(z0, z1))
U32'(active(z0), z1) → c73(U32'(z0, z1))
U32'(z0, active(z1)) → c74(U32'(z0, z1))
U33'(mark(z0)) → c75(U33'(z0))
U33'(active(z0)) → c76(U33'(z0))
U41'(mark(z0), z1) → c77(U41'(z0, z1))
U41'(z0, mark(z1)) → c78(U41'(z0, z1))
U41'(active(z0), z1) → c79(U41'(z0, z1))
U41'(z0, active(z1)) → c80(U41'(z0, z1))
U51'(mark(z0), z1, z2) → c81(U51'(z0, z1, z2))
U51'(z0, mark(z1), z2) → c82(U51'(z0, z1, z2))
U51'(z0, z1, mark(z2)) → c83(U51'(z0, z1, z2))
U51'(active(z0), z1, z2) → c84(U51'(z0, z1, z2))
U51'(z0, active(z1), z2) → c85(U51'(z0, z1, z2))
U51'(z0, z1, active(z2)) → c86(U51'(z0, z1, z2))
S(mark(z0)) → c87(S(z0))
S(active(z0)) → c88(S(z0))
PLUS(mark(z0), z1) → c89(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c90(PLUS(z0, z1))
PLUS(active(z0), z1) → c91(PLUS(z0, z1))
PLUS(z0, active(z1)) → c92(PLUS(z0, z1))
U61'(mark(z0)) → c93(U61'(z0))
U61'(active(z0)) → c94(U61'(z0))
U71'(mark(z0), z1, z2) → c95(U71'(z0, z1, z2))
U71'(z0, mark(z1), z2) → c96(U71'(z0, z1, z2))
U71'(z0, z1, mark(z2)) → c97(U71'(z0, z1, z2))
U71'(active(z0), z1, z2) → c98(U71'(z0, z1, z2))
U71'(z0, active(z1), z2) → c99(U71'(z0, z1, z2))
U71'(z0, z1, active(z2)) → c100(U71'(z0, z1, z2))
X(mark(z0), z1) → c101(X(z0, z1))
X(z0, mark(z1)) → c102(X(z0, z1))
X(active(z0), z1) → c103(X(z0, z1))
X(z0, active(z1)) → c104(X(z0, z1))
AND(mark(z0), z1) → c105(AND(z0, z1))
AND(z0, mark(z1)) → c106(AND(z0, z1))
AND(active(z0), z1) → c107(AND(z0, z1))
AND(z0, active(z1)) → c108(AND(z0, z1))
ISNATKIND(mark(z0)) → c109(ISNATKIND(z0))
ISNATKIND(active(z0)) → c110(ISNATKIND(z0))

K tuples:none
Defined Rule Symbols:

active, mark, U11, U12, isNat, U13, U21, U22, U31, U32, U33, U41, U51, s, plus, U61, U71, x, and, isNatKind

Defined Pair Symbols:

ACTIVE, MARK, U11', U12', ISNAT, U13', U21', U22', U31', U32', U33', U41', U51', S, PLUS, U61', U71', X, AND, ISNATKIND

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c70, c71, c72, c73, c74, c75, c76, c77, c78, c79, c80, c81, c82, c83, c84, c85, c86, c87, c88, c89, c90, c91, c92, c93, c94, c95, c96, c97, c98, c99, c100, c101, c102, c103, c104, c105, c106, c107, c108, c109, c110

(3) CdtUnreachableProof (EQUIVALENT transformation)

The following tuples could be removed as they are not reachable from basic start terms:

ACTIVE(U11(tt, z0, z1)) → c(MARK(U12(isNat(z0), z1)), U12'(isNat(z0), z1), ISNAT(z0))
ACTIVE(U12(tt, z0)) → c1(MARK(U13(isNat(z0))), U13'(isNat(z0)), ISNAT(z0))
ACTIVE(U13(tt)) → c2(MARK(tt))
ACTIVE(U21(tt, z0)) → c3(MARK(U22(isNat(z0))), U22'(isNat(z0)), ISNAT(z0))
ACTIVE(U22(tt)) → c4(MARK(tt))
ACTIVE(U31(tt, z0, z1)) → c5(MARK(U32(isNat(z0), z1)), U32'(isNat(z0), z1), ISNAT(z0))
ACTIVE(U32(tt, z0)) → c6(MARK(U33(isNat(z0))), U33'(isNat(z0)), ISNAT(z0))
ACTIVE(U33(tt)) → c7(MARK(tt))
ACTIVE(U41(tt, z0)) → c8(MARK(z0))
ACTIVE(U51(tt, z0, z1)) → c9(MARK(s(plus(z1, z0))), S(plus(z1, z0)), PLUS(z1, z0))
ACTIVE(U61(tt)) → c10(MARK(0))
ACTIVE(U71(tt, z0, z1)) → c11(MARK(plus(x(z1, z0), z1)), PLUS(x(z1, z0), z1), X(z1, z0))
ACTIVE(and(tt, z0)) → c12(MARK(z0))
ACTIVE(isNat(0)) → c13(MARK(tt))
ACTIVE(isNat(plus(z0, z1))) → c14(MARK(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1)), U11'(and(isNatKind(z0), isNatKind(z1)), z0, z1), AND(isNatKind(z0), isNatKind(z1)), ISNATKIND(z0), ISNATKIND(z1))
ACTIVE(isNat(s(z0))) → c15(MARK(U21(isNatKind(z0), z0)), U21'(isNatKind(z0), z0), ISNATKIND(z0))
ACTIVE(isNat(x(z0, z1))) → c16(MARK(U31(and(isNatKind(z0), isNatKind(z1)), z0, z1)), U31'(and(isNatKind(z0), isNatKind(z1)), z0, z1), AND(isNatKind(z0), isNatKind(z1)), ISNATKIND(z0), ISNATKIND(z1))
ACTIVE(isNatKind(0)) → c17(MARK(tt))
ACTIVE(isNatKind(plus(z0, z1))) → c18(MARK(and(isNatKind(z0), isNatKind(z1))), AND(isNatKind(z0), isNatKind(z1)), ISNATKIND(z0), ISNATKIND(z1))
ACTIVE(isNatKind(s(z0))) → c19(MARK(isNatKind(z0)), ISNATKIND(z0))
ACTIVE(isNatKind(x(z0, z1))) → c20(MARK(and(isNatKind(z0), isNatKind(z1))), AND(isNatKind(z0), isNatKind(z1)), ISNATKIND(z0), ISNATKIND(z1))
ACTIVE(plus(z0, 0)) → c21(MARK(U41(and(isNat(z0), isNatKind(z0)), z0)), U41'(and(isNat(z0), isNatKind(z0)), z0), AND(isNat(z0), isNatKind(z0)), ISNAT(z0), ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c22(MARK(U51(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), U51'(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0), AND(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), AND(isNat(z1), isNatKind(z1)), ISNAT(z1), ISNATKIND(z1), AND(isNat(z0), isNatKind(z0)), ISNAT(z0), ISNATKIND(z0))
ACTIVE(x(z0, 0)) → c23(MARK(U61(and(isNat(z0), isNatKind(z0)))), U61'(and(isNat(z0), isNatKind(z0))), AND(isNat(z0), isNatKind(z0)), ISNAT(z0), ISNATKIND(z0))
ACTIVE(x(z0, s(z1))) → c24(MARK(U71(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), U71'(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0), AND(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), AND(isNat(z1), isNatKind(z1)), ISNAT(z1), ISNATKIND(z1), AND(isNat(z0), isNatKind(z0)), ISNAT(z0), ISNATKIND(z0))
MARK(U11(z0, z1, z2)) → c25(ACTIVE(U11(mark(z0), z1, z2)), U11'(mark(z0), z1, z2), MARK(z0))
MARK(U12(z0, z1)) → c27(ACTIVE(U12(mark(z0), z1)), U12'(mark(z0), z1), MARK(z0))
MARK(isNat(z0)) → c28(ACTIVE(isNat(z0)), ISNAT(z0))
MARK(U13(z0)) → c29(ACTIVE(U13(mark(z0))), U13'(mark(z0)), MARK(z0))
MARK(U21(z0, z1)) → c30(ACTIVE(U21(mark(z0), z1)), U21'(mark(z0), z1), MARK(z0))
MARK(U22(z0)) → c31(ACTIVE(U22(mark(z0))), U22'(mark(z0)), MARK(z0))
MARK(U31(z0, z1, z2)) → c32(ACTIVE(U31(mark(z0), z1, z2)), U31'(mark(z0), z1, z2), MARK(z0))
MARK(U32(z0, z1)) → c33(ACTIVE(U32(mark(z0), z1)), U32'(mark(z0), z1), MARK(z0))
MARK(U33(z0)) → c34(ACTIVE(U33(mark(z0))), U33'(mark(z0)), MARK(z0))
MARK(U41(z0, z1)) → c35(ACTIVE(U41(mark(z0), z1)), U41'(mark(z0), z1), MARK(z0))
MARK(U51(z0, z1, z2)) → c36(ACTIVE(U51(mark(z0), z1, z2)), U51'(mark(z0), z1, z2), MARK(z0))
MARK(s(z0)) → c37(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(plus(z0, z1)) → c38(ACTIVE(plus(mark(z0), mark(z1))), PLUS(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(U61(z0)) → c39(ACTIVE(U61(mark(z0))), U61'(mark(z0)), MARK(z0))
MARK(U71(z0, z1, z2)) → c41(ACTIVE(U71(mark(z0), z1, z2)), U71'(mark(z0), z1, z2), MARK(z0))
MARK(x(z0, z1)) → c42(ACTIVE(x(mark(z0), mark(z1))), X(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(and(z0, z1)) → c43(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(isNatKind(z0)) → c44(ACTIVE(isNatKind(z0)), ISNATKIND(z0))
U11'(mark(z0), z1, z2) → c45(U11'(z0, z1, z2))
U11'(z0, mark(z1), z2) → c46(U11'(z0, z1, z2))
U11'(z0, z1, mark(z2)) → c47(U11'(z0, z1, z2))
U11'(active(z0), z1, z2) → c48(U11'(z0, z1, z2))
U11'(z0, active(z1), z2) → c49(U11'(z0, z1, z2))
U11'(z0, z1, active(z2)) → c50(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c51(U12'(z0, z1))
U12'(z0, mark(z1)) → c52(U12'(z0, z1))
U12'(active(z0), z1) → c53(U12'(z0, z1))
U12'(z0, active(z1)) → c54(U12'(z0, z1))
ISNAT(mark(z0)) → c55(ISNAT(z0))
ISNAT(active(z0)) → c56(ISNAT(z0))
U13'(mark(z0)) → c57(U13'(z0))
U13'(active(z0)) → c58(U13'(z0))
U21'(mark(z0), z1) → c59(U21'(z0, z1))
U21'(z0, mark(z1)) → c60(U21'(z0, z1))
U21'(active(z0), z1) → c61(U21'(z0, z1))
U21'(z0, active(z1)) → c62(U21'(z0, z1))
U22'(mark(z0)) → c63(U22'(z0))
U22'(active(z0)) → c64(U22'(z0))
U31'(mark(z0), z1, z2) → c65(U31'(z0, z1, z2))
U31'(z0, mark(z1), z2) → c66(U31'(z0, z1, z2))
U31'(z0, z1, mark(z2)) → c67(U31'(z0, z1, z2))
U31'(active(z0), z1, z2) → c68(U31'(z0, z1, z2))
U31'(z0, active(z1), z2) → c69(U31'(z0, z1, z2))
U31'(z0, z1, active(z2)) → c70(U31'(z0, z1, z2))
U32'(mark(z0), z1) → c71(U32'(z0, z1))
U32'(z0, mark(z1)) → c72(U32'(z0, z1))
U32'(active(z0), z1) → c73(U32'(z0, z1))
U32'(z0, active(z1)) → c74(U32'(z0, z1))
U33'(mark(z0)) → c75(U33'(z0))
U33'(active(z0)) → c76(U33'(z0))
U41'(mark(z0), z1) → c77(U41'(z0, z1))
U41'(z0, mark(z1)) → c78(U41'(z0, z1))
U41'(active(z0), z1) → c79(U41'(z0, z1))
U41'(z0, active(z1)) → c80(U41'(z0, z1))
U51'(mark(z0), z1, z2) → c81(U51'(z0, z1, z2))
U51'(z0, mark(z1), z2) → c82(U51'(z0, z1, z2))
U51'(z0, z1, mark(z2)) → c83(U51'(z0, z1, z2))
U51'(active(z0), z1, z2) → c84(U51'(z0, z1, z2))
U51'(z0, active(z1), z2) → c85(U51'(z0, z1, z2))
U51'(z0, z1, active(z2)) → c86(U51'(z0, z1, z2))
S(mark(z0)) → c87(S(z0))
S(active(z0)) → c88(S(z0))
PLUS(mark(z0), z1) → c89(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c90(PLUS(z0, z1))
PLUS(active(z0), z1) → c91(PLUS(z0, z1))
PLUS(z0, active(z1)) → c92(PLUS(z0, z1))
U61'(mark(z0)) → c93(U61'(z0))
U61'(active(z0)) → c94(U61'(z0))
U71'(mark(z0), z1, z2) → c95(U71'(z0, z1, z2))
U71'(z0, mark(z1), z2) → c96(U71'(z0, z1, z2))
U71'(z0, z1, mark(z2)) → c97(U71'(z0, z1, z2))
U71'(active(z0), z1, z2) → c98(U71'(z0, z1, z2))
U71'(z0, active(z1), z2) → c99(U71'(z0, z1, z2))
U71'(z0, z1, active(z2)) → c100(U71'(z0, z1, z2))
X(mark(z0), z1) → c101(X(z0, z1))
X(z0, mark(z1)) → c102(X(z0, z1))
X(active(z0), z1) → c103(X(z0, z1))
X(z0, active(z1)) → c104(X(z0, z1))
AND(mark(z0), z1) → c105(AND(z0, z1))
AND(z0, mark(z1)) → c106(AND(z0, z1))
AND(active(z0), z1) → c107(AND(z0, z1))
AND(z0, active(z1)) → c108(AND(z0, z1))
ISNATKIND(mark(z0)) → c109(ISNATKIND(z0))
ISNATKIND(active(z0)) → c110(ISNATKIND(z0))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0, z1)) → mark(U32(isNat(z0), z1))
active(U32(tt, z0)) → mark(U33(isNat(z0)))
active(U33(tt)) → mark(tt)
active(U41(tt, z0)) → mark(z0)
active(U51(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U61(tt)) → mark(0)
active(U71(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNat(x(z0, z1))) → mark(U31(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(isNatKind(x(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(plus(z0, 0)) → mark(U41(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U51(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(x(z0, 0)) → mark(U61(and(isNat(z0), isNatKind(z0))))
active(x(z0, s(z1))) → mark(U71(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
mark(U11(z0, z1, z2)) → active(U11(mark(z0), z1, z2))
mark(tt) → active(tt)
mark(U12(z0, z1)) → active(U12(mark(z0), z1))
mark(isNat(z0)) → active(isNat(z0))
mark(U13(z0)) → active(U13(mark(z0)))
mark(U21(z0, z1)) → active(U21(mark(z0), z1))
mark(U22(z0)) → active(U22(mark(z0)))
mark(U31(z0, z1, z2)) → active(U31(mark(z0), z1, z2))
mark(U32(z0, z1)) → active(U32(mark(z0), z1))
mark(U33(z0)) → active(U33(mark(z0)))
mark(U41(z0, z1)) → active(U41(mark(z0), z1))
mark(U51(z0, z1, z2)) → active(U51(mark(z0), z1, z2))
mark(s(z0)) → active(s(mark(z0)))
mark(plus(z0, z1)) → active(plus(mark(z0), mark(z1)))
mark(U61(z0)) → active(U61(mark(z0)))
mark(0) → active(0)
mark(U71(z0, z1, z2)) → active(U71(mark(z0), z1, z2))
mark(x(z0, z1)) → active(x(mark(z0), mark(z1)))
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(isNatKind(z0)) → active(isNatKind(z0))
U11(mark(z0), z1, z2) → U11(z0, z1, z2)
U11(z0, mark(z1), z2) → U11(z0, z1, z2)
U11(z0, z1, mark(z2)) → U11(z0, z1, z2)
U11(active(z0), z1, z2) → U11(z0, z1, z2)
U11(z0, active(z1), z2) → U11(z0, z1, z2)
U11(z0, z1, active(z2)) → U11(z0, z1, z2)
U12(mark(z0), z1) → U12(z0, z1)
U12(z0, mark(z1)) → U12(z0, z1)
U12(active(z0), z1) → U12(z0, z1)
U12(z0, active(z1)) → U12(z0, z1)
isNat(mark(z0)) → isNat(z0)
isNat(active(z0)) → isNat(z0)
U13(mark(z0)) → U13(z0)
U13(active(z0)) → U13(z0)
U21(mark(z0), z1) → U21(z0, z1)
U21(z0, mark(z1)) → U21(z0, z1)
U21(active(z0), z1) → U21(z0, z1)
U21(z0, active(z1)) → U21(z0, z1)
U22(mark(z0)) → U22(z0)
U22(active(z0)) → U22(z0)
U31(mark(z0), z1, z2) → U31(z0, z1, z2)
U31(z0, mark(z1), z2) → U31(z0, z1, z2)
U31(z0, z1, mark(z2)) → U31(z0, z1, z2)
U31(active(z0), z1, z2) → U31(z0, z1, z2)
U31(z0, active(z1), z2) → U31(z0, z1, z2)
U31(z0, z1, active(z2)) → U31(z0, z1, z2)
U32(mark(z0), z1) → U32(z0, z1)
U32(z0, mark(z1)) → U32(z0, z1)
U32(active(z0), z1) → U32(z0, z1)
U32(z0, active(z1)) → U32(z0, z1)
U33(mark(z0)) → U33(z0)
U33(active(z0)) → U33(z0)
U41(mark(z0), z1) → U41(z0, z1)
U41(z0, mark(z1)) → U41(z0, z1)
U41(active(z0), z1) → U41(z0, z1)
U41(z0, active(z1)) → U41(z0, z1)
U51(mark(z0), z1, z2) → U51(z0, z1, z2)
U51(z0, mark(z1), z2) → U51(z0, z1, z2)
U51(z0, z1, mark(z2)) → U51(z0, z1, z2)
U51(active(z0), z1, z2) → U51(z0, z1, z2)
U51(z0, active(z1), z2) → U51(z0, z1, z2)
U51(z0, z1, active(z2)) → U51(z0, z1, z2)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
plus(mark(z0), z1) → plus(z0, z1)
plus(z0, mark(z1)) → plus(z0, z1)
plus(active(z0), z1) → plus(z0, z1)
plus(z0, active(z1)) → plus(z0, z1)
U61(mark(z0)) → U61(z0)
U61(active(z0)) → U61(z0)
U71(mark(z0), z1, z2) → U71(z0, z1, z2)
U71(z0, mark(z1), z2) → U71(z0, z1, z2)
U71(z0, z1, mark(z2)) → U71(z0, z1, z2)
U71(active(z0), z1, z2) → U71(z0, z1, z2)
U71(z0, active(z1), z2) → U71(z0, z1, z2)
U71(z0, z1, active(z2)) → U71(z0, z1, z2)
x(mark(z0), z1) → x(z0, z1)
x(z0, mark(z1)) → x(z0, z1)
x(active(z0), z1) → x(z0, z1)
x(z0, active(z1)) → x(z0, z1)
and(mark(z0), z1) → and(z0, z1)
and(z0, mark(z1)) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
isNatKind(mark(z0)) → isNatKind(z0)
isNatKind(active(z0)) → isNatKind(z0)

Tuples:

MARK(tt) → c26(ACTIVE(tt))
MARK(0) → c40(ACTIVE(0))

S tuples:

MARK(tt) → c26(ACTIVE(tt))
MARK(0) → c40(ACTIVE(0))

K tuples:none
Defined Rule Symbols:

active, mark, U11, U12, isNat, U13, U21, U22, U31, U32, U33, U41, U51, s, plus, U61, U71, x, and, isNatKind

Defined Pair Symbols:

MARK

Compound Symbols:

c26, c40

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

Removed 2 trailing nodes:

MARK(tt) → c26(ACTIVE(tt))
MARK(0) → c40(ACTIVE(0))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0, z1)) → mark(U12(isNat(z0), z1))
active(U12(tt, z0)) → mark(U13(isNat(z0)))
active(U13(tt)) → mark(tt)
active(U21(tt, z0)) → mark(U22(isNat(z0)))
active(U22(tt)) → mark(tt)
active(U31(tt, z0, z1)) → mark(U32(isNat(z0), z1))
active(U32(tt, z0)) → mark(U33(isNat(z0)))
active(U33(tt)) → mark(tt)
active(U41(tt, z0)) → mark(z0)
active(U51(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U61(tt)) → mark(0)
active(U71(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNat(s(z0))) → mark(U21(isNatKind(z0), z0))
active(isNat(x(z0, z1))) → mark(U31(and(isNatKind(z0), isNatKind(z1)), z0, z1))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(isNatKind(x(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(plus(z0, 0)) → mark(U41(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U51(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
active(x(z0, 0)) → mark(U61(and(isNat(z0), isNatKind(z0))))
active(x(z0, s(z1))) → mark(U71(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0))
mark(U11(z0, z1, z2)) → active(U11(mark(z0), z1, z2))
mark(tt) → active(tt)
mark(U12(z0, z1)) → active(U12(mark(z0), z1))
mark(isNat(z0)) → active(isNat(z0))
mark(U13(z0)) → active(U13(mark(z0)))
mark(U21(z0, z1)) → active(U21(mark(z0), z1))
mark(U22(z0)) → active(U22(mark(z0)))
mark(U31(z0, z1, z2)) → active(U31(mark(z0), z1, z2))
mark(U32(z0, z1)) → active(U32(mark(z0), z1))
mark(U33(z0)) → active(U33(mark(z0)))
mark(U41(z0, z1)) → active(U41(mark(z0), z1))
mark(U51(z0, z1, z2)) → active(U51(mark(z0), z1, z2))
mark(s(z0)) → active(s(mark(z0)))
mark(plus(z0, z1)) → active(plus(mark(z0), mark(z1)))
mark(U61(z0)) → active(U61(mark(z0)))
mark(0) → active(0)
mark(U71(z0, z1, z2)) → active(U71(mark(z0), z1, z2))
mark(x(z0, z1)) → active(x(mark(z0), mark(z1)))
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(isNatKind(z0)) → active(isNatKind(z0))
U11(mark(z0), z1, z2) → U11(z0, z1, z2)
U11(z0, mark(z1), z2) → U11(z0, z1, z2)
U11(z0, z1, mark(z2)) → U11(z0, z1, z2)
U11(active(z0), z1, z2) → U11(z0, z1, z2)
U11(z0, active(z1), z2) → U11(z0, z1, z2)
U11(z0, z1, active(z2)) → U11(z0, z1, z2)
U12(mark(z0), z1) → U12(z0, z1)
U12(z0, mark(z1)) → U12(z0, z1)
U12(active(z0), z1) → U12(z0, z1)
U12(z0, active(z1)) → U12(z0, z1)
isNat(mark(z0)) → isNat(z0)
isNat(active(z0)) → isNat(z0)
U13(mark(z0)) → U13(z0)
U13(active(z0)) → U13(z0)
U21(mark(z0), z1) → U21(z0, z1)
U21(z0, mark(z1)) → U21(z0, z1)
U21(active(z0), z1) → U21(z0, z1)
U21(z0, active(z1)) → U21(z0, z1)
U22(mark(z0)) → U22(z0)
U22(active(z0)) → U22(z0)
U31(mark(z0), z1, z2) → U31(z0, z1, z2)
U31(z0, mark(z1), z2) → U31(z0, z1, z2)
U31(z0, z1, mark(z2)) → U31(z0, z1, z2)
U31(active(z0), z1, z2) → U31(z0, z1, z2)
U31(z0, active(z1), z2) → U31(z0, z1, z2)
U31(z0, z1, active(z2)) → U31(z0, z1, z2)
U32(mark(z0), z1) → U32(z0, z1)
U32(z0, mark(z1)) → U32(z0, z1)
U32(active(z0), z1) → U32(z0, z1)
U32(z0, active(z1)) → U32(z0, z1)
U33(mark(z0)) → U33(z0)
U33(active(z0)) → U33(z0)
U41(mark(z0), z1) → U41(z0, z1)
U41(z0, mark(z1)) → U41(z0, z1)
U41(active(z0), z1) → U41(z0, z1)
U41(z0, active(z1)) → U41(z0, z1)
U51(mark(z0), z1, z2) → U51(z0, z1, z2)
U51(z0, mark(z1), z2) → U51(z0, z1, z2)
U51(z0, z1, mark(z2)) → U51(z0, z1, z2)
U51(active(z0), z1, z2) → U51(z0, z1, z2)
U51(z0, active(z1), z2) → U51(z0, z1, z2)
U51(z0, z1, active(z2)) → U51(z0, z1, z2)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
plus(mark(z0), z1) → plus(z0, z1)
plus(z0, mark(z1)) → plus(z0, z1)
plus(active(z0), z1) → plus(z0, z1)
plus(z0, active(z1)) → plus(z0, z1)
U61(mark(z0)) → U61(z0)
U61(active(z0)) → U61(z0)
U71(mark(z0), z1, z2) → U71(z0, z1, z2)
U71(z0, mark(z1), z2) → U71(z0, z1, z2)
U71(z0, z1, mark(z2)) → U71(z0, z1, z2)
U71(active(z0), z1, z2) → U71(z0, z1, z2)
U71(z0, active(z1), z2) → U71(z0, z1, z2)
U71(z0, z1, active(z2)) → U71(z0, z1, z2)
x(mark(z0), z1) → x(z0, z1)
x(z0, mark(z1)) → x(z0, z1)
x(active(z0), z1) → x(z0, z1)
x(z0, active(z1)) → x(z0, z1)
and(mark(z0), z1) → and(z0, z1)
and(z0, mark(z1)) → and(z0, z1)
and(active(z0), z1) → and(z0, z1)
and(z0, active(z1)) → and(z0, z1)
isNatKind(mark(z0)) → isNatKind(z0)
isNatKind(active(z0)) → isNatKind(z0)

Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

active, mark, U11, U12, isNat, U13, U21, U22, U31, U32, U33, U41, U51, s, plus, U61, U71, x, and, isNatKind

Defined Pair Symbols:none

Compound Symbols:none

(7) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(8) BOUNDS(O(1), O(1))