(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, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0)) → mark(U31(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U41(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)) → active(U31(mark(X1), X2))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(0) → active(0)
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) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(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)
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)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(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)) → active(U31(mark(z0), z1))
mark(U41(z0, z1, z2)) → active(U41(mark(z0), z1, z2))
mark(s(z0)) → active(s(mark(z0)))
mark(plus(z0, z1)) → active(plus(mark(z0), mark(z1)))
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(0) → active(0)
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) → U31(z0, z1)
U31(z0, mark(z1)) → U31(z0, z1)
U31(active(z0), z1) → U31(z0, z1)
U31(z0, active(z1)) → U31(z0, z1)
U41(mark(z0), z1, z2) → U41(z0, z1, z2)
U41(z0, mark(z1), z2) → U41(z0, z1, z2)
U41(z0, z1, mark(z2)) → U41(z0, z1, z2)
U41(active(z0), z1, z2) → U41(z0, z1, z2)
U41(z0, active(z1), z2) → U41(z0, z1, z2)
U41(z0, z1, active(z2)) → U41(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)
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)) → c5(MARK(z0))
ACTIVE(U41(tt, z0, z1)) → c6(MARK(s(plus(z1, z0))), S(plus(z1, z0)), PLUS(z1, z0))
ACTIVE(and(tt, z0)) → c7(MARK(z0))
ACTIVE(isNat(0)) → c8(MARK(tt))
ACTIVE(isNat(plus(z0, z1))) → c9(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))) → c10(MARK(U21(isNatKind(z0), z0)), U21'(isNatKind(z0), z0), ISNATKIND(z0))
ACTIVE(isNatKind(0)) → c11(MARK(tt))
ACTIVE(isNatKind(plus(z0, z1))) → c12(MARK(and(isNatKind(z0), isNatKind(z1))), AND(isNatKind(z0), isNatKind(z1)), ISNATKIND(z0), ISNATKIND(z1))
ACTIVE(isNatKind(s(z0))) → c13(MARK(isNatKind(z0)), ISNATKIND(z0))
ACTIVE(plus(z0, 0)) → c14(MARK(U31(and(isNat(z0), isNatKind(z0)), z0)), U31'(and(isNat(z0), isNatKind(z0)), z0), AND(isNat(z0), isNatKind(z0)), ISNAT(z0), ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c15(MARK(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), U41'(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)) → c16(ACTIVE(U11(mark(z0), z1, z2)), U11'(mark(z0), z1, z2), MARK(z0))
MARK(tt) → c17(ACTIVE(tt))
MARK(U12(z0, z1)) → c18(ACTIVE(U12(mark(z0), z1)), U12'(mark(z0), z1), MARK(z0))
MARK(isNat(z0)) → c19(ACTIVE(isNat(z0)), ISNAT(z0))
MARK(U13(z0)) → c20(ACTIVE(U13(mark(z0))), U13'(mark(z0)), MARK(z0))
MARK(U21(z0, z1)) → c21(ACTIVE(U21(mark(z0), z1)), U21'(mark(z0), z1), MARK(z0))
MARK(U22(z0)) → c22(ACTIVE(U22(mark(z0))), U22'(mark(z0)), MARK(z0))
MARK(U31(z0, z1)) → c23(ACTIVE(U31(mark(z0), z1)), U31'(mark(z0), z1), MARK(z0))
MARK(U41(z0, z1, z2)) → c24(ACTIVE(U41(mark(z0), z1, z2)), U41'(mark(z0), z1, z2), MARK(z0))
MARK(s(z0)) → c25(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(plus(z0, z1)) → c26(ACTIVE(plus(mark(z0), mark(z1))), PLUS(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(and(z0, z1)) → c27(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(0) → c28(ACTIVE(0))
MARK(isNatKind(z0)) → c29(ACTIVE(isNatKind(z0)), ISNATKIND(z0))
U11'(mark(z0), z1, z2) → c30(U11'(z0, z1, z2))
U11'(z0, mark(z1), z2) → c31(U11'(z0, z1, z2))
U11'(z0, z1, mark(z2)) → c32(U11'(z0, z1, z2))
U11'(active(z0), z1, z2) → c33(U11'(z0, z1, z2))
U11'(z0, active(z1), z2) → c34(U11'(z0, z1, z2))
U11'(z0, z1, active(z2)) → c35(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c36(U12'(z0, z1))
U12'(z0, mark(z1)) → c37(U12'(z0, z1))
U12'(active(z0), z1) → c38(U12'(z0, z1))
U12'(z0, active(z1)) → c39(U12'(z0, z1))
ISNAT(mark(z0)) → c40(ISNAT(z0))
ISNAT(active(z0)) → c41(ISNAT(z0))
U13'(mark(z0)) → c42(U13'(z0))
U13'(active(z0)) → c43(U13'(z0))
U21'(mark(z0), z1) → c44(U21'(z0, z1))
U21'(z0, mark(z1)) → c45(U21'(z0, z1))
U21'(active(z0), z1) → c46(U21'(z0, z1))
U21'(z0, active(z1)) → c47(U21'(z0, z1))
U22'(mark(z0)) → c48(U22'(z0))
U22'(active(z0)) → c49(U22'(z0))
U31'(mark(z0), z1) → c50(U31'(z0, z1))
U31'(z0, mark(z1)) → c51(U31'(z0, z1))
U31'(active(z0), z1) → c52(U31'(z0, z1))
U31'(z0, active(z1)) → c53(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c54(U41'(z0, z1, z2))
U41'(z0, mark(z1), z2) → c55(U41'(z0, z1, z2))
U41'(z0, z1, mark(z2)) → c56(U41'(z0, z1, z2))
U41'(active(z0), z1, z2) → c57(U41'(z0, z1, z2))
U41'(z0, active(z1), z2) → c58(U41'(z0, z1, z2))
U41'(z0, z1, active(z2)) → c59(U41'(z0, z1, z2))
S(mark(z0)) → c60(S(z0))
S(active(z0)) → c61(S(z0))
PLUS(mark(z0), z1) → c62(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c63(PLUS(z0, z1))
PLUS(active(z0), z1) → c64(PLUS(z0, z1))
PLUS(z0, active(z1)) → c65(PLUS(z0, z1))
AND(mark(z0), z1) → c66(AND(z0, z1))
AND(z0, mark(z1)) → c67(AND(z0, z1))
AND(active(z0), z1) → c68(AND(z0, z1))
AND(z0, active(z1)) → c69(AND(z0, z1))
ISNATKIND(mark(z0)) → c70(ISNATKIND(z0))
ISNATKIND(active(z0)) → c71(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)) → c5(MARK(z0))
ACTIVE(U41(tt, z0, z1)) → c6(MARK(s(plus(z1, z0))), S(plus(z1, z0)), PLUS(z1, z0))
ACTIVE(and(tt, z0)) → c7(MARK(z0))
ACTIVE(isNat(0)) → c8(MARK(tt))
ACTIVE(isNat(plus(z0, z1))) → c9(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))) → c10(MARK(U21(isNatKind(z0), z0)), U21'(isNatKind(z0), z0), ISNATKIND(z0))
ACTIVE(isNatKind(0)) → c11(MARK(tt))
ACTIVE(isNatKind(plus(z0, z1))) → c12(MARK(and(isNatKind(z0), isNatKind(z1))), AND(isNatKind(z0), isNatKind(z1)), ISNATKIND(z0), ISNATKIND(z1))
ACTIVE(isNatKind(s(z0))) → c13(MARK(isNatKind(z0)), ISNATKIND(z0))
ACTIVE(plus(z0, 0)) → c14(MARK(U31(and(isNat(z0), isNatKind(z0)), z0)), U31'(and(isNat(z0), isNatKind(z0)), z0), AND(isNat(z0), isNatKind(z0)), ISNAT(z0), ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c15(MARK(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), U41'(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)) → c16(ACTIVE(U11(mark(z0), z1, z2)), U11'(mark(z0), z1, z2), MARK(z0))
MARK(tt) → c17(ACTIVE(tt))
MARK(U12(z0, z1)) → c18(ACTIVE(U12(mark(z0), z1)), U12'(mark(z0), z1), MARK(z0))
MARK(isNat(z0)) → c19(ACTIVE(isNat(z0)), ISNAT(z0))
MARK(U13(z0)) → c20(ACTIVE(U13(mark(z0))), U13'(mark(z0)), MARK(z0))
MARK(U21(z0, z1)) → c21(ACTIVE(U21(mark(z0), z1)), U21'(mark(z0), z1), MARK(z0))
MARK(U22(z0)) → c22(ACTIVE(U22(mark(z0))), U22'(mark(z0)), MARK(z0))
MARK(U31(z0, z1)) → c23(ACTIVE(U31(mark(z0), z1)), U31'(mark(z0), z1), MARK(z0))
MARK(U41(z0, z1, z2)) → c24(ACTIVE(U41(mark(z0), z1, z2)), U41'(mark(z0), z1, z2), MARK(z0))
MARK(s(z0)) → c25(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(plus(z0, z1)) → c26(ACTIVE(plus(mark(z0), mark(z1))), PLUS(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(and(z0, z1)) → c27(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(0) → c28(ACTIVE(0))
MARK(isNatKind(z0)) → c29(ACTIVE(isNatKind(z0)), ISNATKIND(z0))
U11'(mark(z0), z1, z2) → c30(U11'(z0, z1, z2))
U11'(z0, mark(z1), z2) → c31(U11'(z0, z1, z2))
U11'(z0, z1, mark(z2)) → c32(U11'(z0, z1, z2))
U11'(active(z0), z1, z2) → c33(U11'(z0, z1, z2))
U11'(z0, active(z1), z2) → c34(U11'(z0, z1, z2))
U11'(z0, z1, active(z2)) → c35(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c36(U12'(z0, z1))
U12'(z0, mark(z1)) → c37(U12'(z0, z1))
U12'(active(z0), z1) → c38(U12'(z0, z1))
U12'(z0, active(z1)) → c39(U12'(z0, z1))
ISNAT(mark(z0)) → c40(ISNAT(z0))
ISNAT(active(z0)) → c41(ISNAT(z0))
U13'(mark(z0)) → c42(U13'(z0))
U13'(active(z0)) → c43(U13'(z0))
U21'(mark(z0), z1) → c44(U21'(z0, z1))
U21'(z0, mark(z1)) → c45(U21'(z0, z1))
U21'(active(z0), z1) → c46(U21'(z0, z1))
U21'(z0, active(z1)) → c47(U21'(z0, z1))
U22'(mark(z0)) → c48(U22'(z0))
U22'(active(z0)) → c49(U22'(z0))
U31'(mark(z0), z1) → c50(U31'(z0, z1))
U31'(z0, mark(z1)) → c51(U31'(z0, z1))
U31'(active(z0), z1) → c52(U31'(z0, z1))
U31'(z0, active(z1)) → c53(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c54(U41'(z0, z1, z2))
U41'(z0, mark(z1), z2) → c55(U41'(z0, z1, z2))
U41'(z0, z1, mark(z2)) → c56(U41'(z0, z1, z2))
U41'(active(z0), z1, z2) → c57(U41'(z0, z1, z2))
U41'(z0, active(z1), z2) → c58(U41'(z0, z1, z2))
U41'(z0, z1, active(z2)) → c59(U41'(z0, z1, z2))
S(mark(z0)) → c60(S(z0))
S(active(z0)) → c61(S(z0))
PLUS(mark(z0), z1) → c62(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c63(PLUS(z0, z1))
PLUS(active(z0), z1) → c64(PLUS(z0, z1))
PLUS(z0, active(z1)) → c65(PLUS(z0, z1))
AND(mark(z0), z1) → c66(AND(z0, z1))
AND(z0, mark(z1)) → c67(AND(z0, z1))
AND(active(z0), z1) → c68(AND(z0, z1))
AND(z0, active(z1)) → c69(AND(z0, z1))
ISNATKIND(mark(z0)) → c70(ISNATKIND(z0))
ISNATKIND(active(z0)) → c71(ISNATKIND(z0))
K tuples:none
Defined Rule Symbols:
active, mark, U11, U12, isNat, U13, U21, U22, U31, U41, s, plus, and, isNatKind
Defined Pair Symbols:
ACTIVE, MARK, U11', U12', ISNAT, U13', U21', U22', U31', U41', S, PLUS, 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
(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)) → c5(MARK(z0))
ACTIVE(U41(tt, z0, z1)) → c6(MARK(s(plus(z1, z0))), S(plus(z1, z0)), PLUS(z1, z0))
ACTIVE(and(tt, z0)) → c7(MARK(z0))
ACTIVE(isNat(0)) → c8(MARK(tt))
ACTIVE(isNat(plus(z0, z1))) → c9(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))) → c10(MARK(U21(isNatKind(z0), z0)), U21'(isNatKind(z0), z0), ISNATKIND(z0))
ACTIVE(isNatKind(0)) → c11(MARK(tt))
ACTIVE(isNatKind(plus(z0, z1))) → c12(MARK(and(isNatKind(z0), isNatKind(z1))), AND(isNatKind(z0), isNatKind(z1)), ISNATKIND(z0), ISNATKIND(z1))
ACTIVE(isNatKind(s(z0))) → c13(MARK(isNatKind(z0)), ISNATKIND(z0))
ACTIVE(plus(z0, 0)) → c14(MARK(U31(and(isNat(z0), isNatKind(z0)), z0)), U31'(and(isNat(z0), isNatKind(z0)), z0), AND(isNat(z0), isNatKind(z0)), ISNAT(z0), ISNATKIND(z0))
ACTIVE(plus(z0, s(z1))) → c15(MARK(U41(and(and(isNat(z1), isNatKind(z1)), and(isNat(z0), isNatKind(z0))), z1, z0)), U41'(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)) → c16(ACTIVE(U11(mark(z0), z1, z2)), U11'(mark(z0), z1, z2), MARK(z0))
MARK(U12(z0, z1)) → c18(ACTIVE(U12(mark(z0), z1)), U12'(mark(z0), z1), MARK(z0))
MARK(isNat(z0)) → c19(ACTIVE(isNat(z0)), ISNAT(z0))
MARK(U13(z0)) → c20(ACTIVE(U13(mark(z0))), U13'(mark(z0)), MARK(z0))
MARK(U21(z0, z1)) → c21(ACTIVE(U21(mark(z0), z1)), U21'(mark(z0), z1), MARK(z0))
MARK(U22(z0)) → c22(ACTIVE(U22(mark(z0))), U22'(mark(z0)), MARK(z0))
MARK(U31(z0, z1)) → c23(ACTIVE(U31(mark(z0), z1)), U31'(mark(z0), z1), MARK(z0))
MARK(U41(z0, z1, z2)) → c24(ACTIVE(U41(mark(z0), z1, z2)), U41'(mark(z0), z1, z2), MARK(z0))
MARK(s(z0)) → c25(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(plus(z0, z1)) → c26(ACTIVE(plus(mark(z0), mark(z1))), PLUS(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(and(z0, z1)) → c27(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(isNatKind(z0)) → c29(ACTIVE(isNatKind(z0)), ISNATKIND(z0))
U11'(mark(z0), z1, z2) → c30(U11'(z0, z1, z2))
U11'(z0, mark(z1), z2) → c31(U11'(z0, z1, z2))
U11'(z0, z1, mark(z2)) → c32(U11'(z0, z1, z2))
U11'(active(z0), z1, z2) → c33(U11'(z0, z1, z2))
U11'(z0, active(z1), z2) → c34(U11'(z0, z1, z2))
U11'(z0, z1, active(z2)) → c35(U11'(z0, z1, z2))
U12'(mark(z0), z1) → c36(U12'(z0, z1))
U12'(z0, mark(z1)) → c37(U12'(z0, z1))
U12'(active(z0), z1) → c38(U12'(z0, z1))
U12'(z0, active(z1)) → c39(U12'(z0, z1))
ISNAT(mark(z0)) → c40(ISNAT(z0))
ISNAT(active(z0)) → c41(ISNAT(z0))
U13'(mark(z0)) → c42(U13'(z0))
U13'(active(z0)) → c43(U13'(z0))
U21'(mark(z0), z1) → c44(U21'(z0, z1))
U21'(z0, mark(z1)) → c45(U21'(z0, z1))
U21'(active(z0), z1) → c46(U21'(z0, z1))
U21'(z0, active(z1)) → c47(U21'(z0, z1))
U22'(mark(z0)) → c48(U22'(z0))
U22'(active(z0)) → c49(U22'(z0))
U31'(mark(z0), z1) → c50(U31'(z0, z1))
U31'(z0, mark(z1)) → c51(U31'(z0, z1))
U31'(active(z0), z1) → c52(U31'(z0, z1))
U31'(z0, active(z1)) → c53(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c54(U41'(z0, z1, z2))
U41'(z0, mark(z1), z2) → c55(U41'(z0, z1, z2))
U41'(z0, z1, mark(z2)) → c56(U41'(z0, z1, z2))
U41'(active(z0), z1, z2) → c57(U41'(z0, z1, z2))
U41'(z0, active(z1), z2) → c58(U41'(z0, z1, z2))
U41'(z0, z1, active(z2)) → c59(U41'(z0, z1, z2))
S(mark(z0)) → c60(S(z0))
S(active(z0)) → c61(S(z0))
PLUS(mark(z0), z1) → c62(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c63(PLUS(z0, z1))
PLUS(active(z0), z1) → c64(PLUS(z0, z1))
PLUS(z0, active(z1)) → c65(PLUS(z0, z1))
AND(mark(z0), z1) → c66(AND(z0, z1))
AND(z0, mark(z1)) → c67(AND(z0, z1))
AND(active(z0), z1) → c68(AND(z0, z1))
AND(z0, active(z1)) → c69(AND(z0, z1))
ISNATKIND(mark(z0)) → c70(ISNATKIND(z0))
ISNATKIND(active(z0)) → c71(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)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(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)) → active(U31(mark(z0), z1))
mark(U41(z0, z1, z2)) → active(U41(mark(z0), z1, z2))
mark(s(z0)) → active(s(mark(z0)))
mark(plus(z0, z1)) → active(plus(mark(z0), mark(z1)))
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(0) → active(0)
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) → U31(z0, z1)
U31(z0, mark(z1)) → U31(z0, z1)
U31(active(z0), z1) → U31(z0, z1)
U31(z0, active(z1)) → U31(z0, z1)
U41(mark(z0), z1, z2) → U41(z0, z1, z2)
U41(z0, mark(z1), z2) → U41(z0, z1, z2)
U41(z0, z1, mark(z2)) → U41(z0, z1, z2)
U41(active(z0), z1, z2) → U41(z0, z1, z2)
U41(z0, active(z1), z2) → U41(z0, z1, z2)
U41(z0, z1, active(z2)) → U41(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)
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) → c17(ACTIVE(tt))
MARK(0) → c28(ACTIVE(0))
S tuples:
MARK(tt) → c17(ACTIVE(tt))
MARK(0) → c28(ACTIVE(0))
K tuples:none
Defined Rule Symbols:
active, mark, U11, U12, isNat, U13, U21, U22, U31, U41, s, plus, and, isNatKind
Defined Pair Symbols:
MARK
Compound Symbols:
c17, c28
(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing nodes:
MARK(tt) → c17(ACTIVE(tt))
MARK(0) → c28(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)) → mark(z0)
active(U41(tt, z0, z1)) → mark(s(plus(z1, z0)))
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(isNatKind(0)) → mark(tt)
active(isNatKind(plus(z0, z1))) → mark(and(isNatKind(z0), isNatKind(z1)))
active(isNatKind(s(z0))) → mark(isNatKind(z0))
active(plus(z0, 0)) → mark(U31(and(isNat(z0), isNatKind(z0)), z0))
active(plus(z0, s(z1))) → mark(U41(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)) → active(U31(mark(z0), z1))
mark(U41(z0, z1, z2)) → active(U41(mark(z0), z1, z2))
mark(s(z0)) → active(s(mark(z0)))
mark(plus(z0, z1)) → active(plus(mark(z0), mark(z1)))
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(0) → active(0)
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) → U31(z0, z1)
U31(z0, mark(z1)) → U31(z0, z1)
U31(active(z0), z1) → U31(z0, z1)
U31(z0, active(z1)) → U31(z0, z1)
U41(mark(z0), z1, z2) → U41(z0, z1, z2)
U41(z0, mark(z1), z2) → U41(z0, z1, z2)
U41(z0, z1, mark(z2)) → U41(z0, z1, z2)
U41(active(z0), z1, z2) → U41(z0, z1, z2)
U41(z0, active(z1), z2) → U41(z0, z1, z2)
U41(z0, z1, active(z2)) → U41(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)
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, U41, s, plus, and, isNatKind
Defined Pair Symbols:none
Compound Symbols:none
(7) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(8) BOUNDS(O(1), O(1))