(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNat(X)) → active(isNat(X))
mark(U21(X)) → active(U21(mark(X)))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X1, X2)) → active(U41(mark(X1), X2))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2, X3)) → active(U52(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(U72(X1, X2, X3)) → active(U72(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U21(mark(X)) → U21(X)
U21(active(X)) → U21(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)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(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)
U52(mark(X1), X2, X3) → U52(X1, X2, X3)
U52(X1, mark(X2), X3) → U52(X1, X2, X3)
U52(X1, X2, mark(X3)) → U52(X1, X2, X3)
U52(active(X1), X2, X3) → U52(X1, X2, X3)
U52(X1, active(X2), X3) → U52(X1, X2, X3)
U52(X1, X2, active(X3)) → U52(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)
U72(mark(X1), X2, X3) → U72(X1, X2, X3)
U72(X1, mark(X2), X3) → U72(X1, X2, X3)
U72(X1, X2, mark(X3)) → U72(X1, X2, X3)
U72(active(X1), X2, X3) → U72(X1, X2, X3)
U72(X1, active(X2), X3) → U72(X1, X2, X3)
U72(X1, X2, active(X3)) → U72(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)
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)) → mark(U12(isNat(z0)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, z0)) → mark(U32(isNat(z0)))
active(U32(tt)) → mark(tt)
active(U41(tt, z0)) → mark(z0)
active(U51(tt, z0, z1)) → mark(U52(isNat(z1), z0, z1))
active(U52(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U61(tt)) → mark(0)
active(U71(tt, z0, z1)) → mark(U72(isNat(z1), z0, z1))
active(U72(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNat(z0), z1))
active(isNat(s(z0))) → mark(U21(isNat(z0)))
active(isNat(x(z0, z1))) → mark(U31(isNat(z0), z1))
active(plus(z0, 0)) → mark(U41(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U51(isNat(z1), z1, z0))
active(x(z0, 0)) → mark(U61(isNat(z0)))
active(x(z0, s(z1))) → mark(U71(isNat(z1), z1, z0))
mark(U11(z0, z1)) → active(U11(mark(z0), z1))
mark(tt) → active(tt)
mark(U12(z0)) → active(U12(mark(z0)))
mark(isNat(z0)) → active(isNat(z0))
mark(U21(z0)) → active(U21(mark(z0)))
mark(U31(z0, z1)) → active(U31(mark(z0), z1))
mark(U32(z0)) → active(U32(mark(z0)))
mark(U41(z0, z1)) → active(U41(mark(z0), z1))
mark(U51(z0, z1, z2)) → active(U51(mark(z0), z1, z2))
mark(U52(z0, z1, z2)) → active(U52(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(U72(z0, z1, z2)) → active(U72(mark(z0), z1, z2))
mark(x(z0, z1)) → active(x(mark(z0), mark(z1)))
U11(mark(z0), z1) → U11(z0, z1)
U11(z0, mark(z1)) → U11(z0, z1)
U11(active(z0), z1) → U11(z0, z1)
U11(z0, active(z1)) → U11(z0, z1)
U12(mark(z0)) → U12(z0)
U12(active(z0)) → U12(z0)
isNat(mark(z0)) → isNat(z0)
isNat(active(z0)) → isNat(z0)
U21(mark(z0)) → U21(z0)
U21(active(z0)) → U21(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)
U32(mark(z0)) → U32(z0)
U32(active(z0)) → U32(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)
U52(mark(z0), z1, z2) → U52(z0, z1, z2)
U52(z0, mark(z1), z2) → U52(z0, z1, z2)
U52(z0, z1, mark(z2)) → U52(z0, z1, z2)
U52(active(z0), z1, z2) → U52(z0, z1, z2)
U52(z0, active(z1), z2) → U52(z0, z1, z2)
U52(z0, z1, active(z2)) → U52(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)
U72(mark(z0), z1, z2) → U72(z0, z1, z2)
U72(z0, mark(z1), z2) → U72(z0, z1, z2)
U72(z0, z1, mark(z2)) → U72(z0, z1, z2)
U72(active(z0), z1, z2) → U72(z0, z1, z2)
U72(z0, active(z1), z2) → U72(z0, z1, z2)
U72(z0, z1, active(z2)) → U72(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)
Tuples:
ACTIVE(U11(tt, z0)) → c(MARK(U12(isNat(z0))), U12'(isNat(z0)), ISNAT(z0))
ACTIVE(U12(tt)) → c1(MARK(tt))
ACTIVE(U21(tt)) → c2(MARK(tt))
ACTIVE(U31(tt, z0)) → c3(MARK(U32(isNat(z0))), U32'(isNat(z0)), ISNAT(z0))
ACTIVE(U32(tt)) → c4(MARK(tt))
ACTIVE(U41(tt, z0)) → c5(MARK(z0))
ACTIVE(U51(tt, z0, z1)) → c6(MARK(U52(isNat(z1), z0, z1)), U52'(isNat(z1), z0, z1), ISNAT(z1))
ACTIVE(U52(tt, z0, z1)) → c7(MARK(s(plus(z1, z0))), S(plus(z1, z0)), PLUS(z1, z0))
ACTIVE(U61(tt)) → c8(MARK(0))
ACTIVE(U71(tt, z0, z1)) → c9(MARK(U72(isNat(z1), z0, z1)), U72'(isNat(z1), z0, z1), ISNAT(z1))
ACTIVE(U72(tt, z0, z1)) → c10(MARK(plus(x(z1, z0), z1)), PLUS(x(z1, z0), z1), X(z1, z0))
ACTIVE(isNat(0)) → c11(MARK(tt))
ACTIVE(isNat(plus(z0, z1))) → c12(MARK(U11(isNat(z0), z1)), U11'(isNat(z0), z1), ISNAT(z0))
ACTIVE(isNat(s(z0))) → c13(MARK(U21(isNat(z0))), U21'(isNat(z0)), ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c14(MARK(U31(isNat(z0), z1)), U31'(isNat(z0), z1), ISNAT(z0))
ACTIVE(plus(z0, 0)) → c15(MARK(U41(isNat(z0), z0)), U41'(isNat(z0), z0), ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c16(MARK(U51(isNat(z1), z1, z0)), U51'(isNat(z1), z1, z0), ISNAT(z1))
ACTIVE(x(z0, 0)) → c17(MARK(U61(isNat(z0))), U61'(isNat(z0)), ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c18(MARK(U71(isNat(z1), z1, z0)), U71'(isNat(z1), z1, z0), ISNAT(z1))
MARK(U11(z0, z1)) → c19(ACTIVE(U11(mark(z0), z1)), U11'(mark(z0), z1), MARK(z0))
MARK(tt) → c20(ACTIVE(tt))
MARK(U12(z0)) → c21(ACTIVE(U12(mark(z0))), U12'(mark(z0)), MARK(z0))
MARK(isNat(z0)) → c22(ACTIVE(isNat(z0)), ISNAT(z0))
MARK(U21(z0)) → c23(ACTIVE(U21(mark(z0))), U21'(mark(z0)), MARK(z0))
MARK(U31(z0, z1)) → c24(ACTIVE(U31(mark(z0), z1)), U31'(mark(z0), z1), MARK(z0))
MARK(U32(z0)) → c25(ACTIVE(U32(mark(z0))), U32'(mark(z0)), MARK(z0))
MARK(U41(z0, z1)) → c26(ACTIVE(U41(mark(z0), z1)), U41'(mark(z0), z1), MARK(z0))
MARK(U51(z0, z1, z2)) → c27(ACTIVE(U51(mark(z0), z1, z2)), U51'(mark(z0), z1, z2), MARK(z0))
MARK(U52(z0, z1, z2)) → c28(ACTIVE(U52(mark(z0), z1, z2)), U52'(mark(z0), z1, z2), MARK(z0))
MARK(s(z0)) → c29(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(plus(z0, z1)) → c30(ACTIVE(plus(mark(z0), mark(z1))), PLUS(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(U61(z0)) → c31(ACTIVE(U61(mark(z0))), U61'(mark(z0)), MARK(z0))
MARK(0) → c32(ACTIVE(0))
MARK(U71(z0, z1, z2)) → c33(ACTIVE(U71(mark(z0), z1, z2)), U71'(mark(z0), z1, z2), MARK(z0))
MARK(U72(z0, z1, z2)) → c34(ACTIVE(U72(mark(z0), z1, z2)), U72'(mark(z0), z1, z2), MARK(z0))
MARK(x(z0, z1)) → c35(ACTIVE(x(mark(z0), mark(z1))), X(mark(z0), mark(z1)), MARK(z0), MARK(z1))
U11'(mark(z0), z1) → c36(U11'(z0, z1))
U11'(z0, mark(z1)) → c37(U11'(z0, z1))
U11'(active(z0), z1) → c38(U11'(z0, z1))
U11'(z0, active(z1)) → c39(U11'(z0, z1))
U12'(mark(z0)) → c40(U12'(z0))
U12'(active(z0)) → c41(U12'(z0))
ISNAT(mark(z0)) → c42(ISNAT(z0))
ISNAT(active(z0)) → c43(ISNAT(z0))
U21'(mark(z0)) → c44(U21'(z0))
U21'(active(z0)) → c45(U21'(z0))
U31'(mark(z0), z1) → c46(U31'(z0, z1))
U31'(z0, mark(z1)) → c47(U31'(z0, z1))
U31'(active(z0), z1) → c48(U31'(z0, z1))
U31'(z0, active(z1)) → c49(U31'(z0, z1))
U32'(mark(z0)) → c50(U32'(z0))
U32'(active(z0)) → c51(U32'(z0))
U41'(mark(z0), z1) → c52(U41'(z0, z1))
U41'(z0, mark(z1)) → c53(U41'(z0, z1))
U41'(active(z0), z1) → c54(U41'(z0, z1))
U41'(z0, active(z1)) → c55(U41'(z0, z1))
U51'(mark(z0), z1, z2) → c56(U51'(z0, z1, z2))
U51'(z0, mark(z1), z2) → c57(U51'(z0, z1, z2))
U51'(z0, z1, mark(z2)) → c58(U51'(z0, z1, z2))
U51'(active(z0), z1, z2) → c59(U51'(z0, z1, z2))
U51'(z0, active(z1), z2) → c60(U51'(z0, z1, z2))
U51'(z0, z1, active(z2)) → c61(U51'(z0, z1, z2))
U52'(mark(z0), z1, z2) → c62(U52'(z0, z1, z2))
U52'(z0, mark(z1), z2) → c63(U52'(z0, z1, z2))
U52'(z0, z1, mark(z2)) → c64(U52'(z0, z1, z2))
U52'(active(z0), z1, z2) → c65(U52'(z0, z1, z2))
U52'(z0, active(z1), z2) → c66(U52'(z0, z1, z2))
U52'(z0, z1, active(z2)) → c67(U52'(z0, z1, z2))
S(mark(z0)) → c68(S(z0))
S(active(z0)) → c69(S(z0))
PLUS(mark(z0), z1) → c70(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c71(PLUS(z0, z1))
PLUS(active(z0), z1) → c72(PLUS(z0, z1))
PLUS(z0, active(z1)) → c73(PLUS(z0, z1))
U61'(mark(z0)) → c74(U61'(z0))
U61'(active(z0)) → c75(U61'(z0))
U71'(mark(z0), z1, z2) → c76(U71'(z0, z1, z2))
U71'(z0, mark(z1), z2) → c77(U71'(z0, z1, z2))
U71'(z0, z1, mark(z2)) → c78(U71'(z0, z1, z2))
U71'(active(z0), z1, z2) → c79(U71'(z0, z1, z2))
U71'(z0, active(z1), z2) → c80(U71'(z0, z1, z2))
U71'(z0, z1, active(z2)) → c81(U71'(z0, z1, z2))
U72'(mark(z0), z1, z2) → c82(U72'(z0, z1, z2))
U72'(z0, mark(z1), z2) → c83(U72'(z0, z1, z2))
U72'(z0, z1, mark(z2)) → c84(U72'(z0, z1, z2))
U72'(active(z0), z1, z2) → c85(U72'(z0, z1, z2))
U72'(z0, active(z1), z2) → c86(U72'(z0, z1, z2))
U72'(z0, z1, active(z2)) → c87(U72'(z0, z1, z2))
X(mark(z0), z1) → c88(X(z0, z1))
X(z0, mark(z1)) → c89(X(z0, z1))
X(active(z0), z1) → c90(X(z0, z1))
X(z0, active(z1)) → c91(X(z0, z1))
S tuples:
ACTIVE(U11(tt, z0)) → c(MARK(U12(isNat(z0))), U12'(isNat(z0)), ISNAT(z0))
ACTIVE(U12(tt)) → c1(MARK(tt))
ACTIVE(U21(tt)) → c2(MARK(tt))
ACTIVE(U31(tt, z0)) → c3(MARK(U32(isNat(z0))), U32'(isNat(z0)), ISNAT(z0))
ACTIVE(U32(tt)) → c4(MARK(tt))
ACTIVE(U41(tt, z0)) → c5(MARK(z0))
ACTIVE(U51(tt, z0, z1)) → c6(MARK(U52(isNat(z1), z0, z1)), U52'(isNat(z1), z0, z1), ISNAT(z1))
ACTIVE(U52(tt, z0, z1)) → c7(MARK(s(plus(z1, z0))), S(plus(z1, z0)), PLUS(z1, z0))
ACTIVE(U61(tt)) → c8(MARK(0))
ACTIVE(U71(tt, z0, z1)) → c9(MARK(U72(isNat(z1), z0, z1)), U72'(isNat(z1), z0, z1), ISNAT(z1))
ACTIVE(U72(tt, z0, z1)) → c10(MARK(plus(x(z1, z0), z1)), PLUS(x(z1, z0), z1), X(z1, z0))
ACTIVE(isNat(0)) → c11(MARK(tt))
ACTIVE(isNat(plus(z0, z1))) → c12(MARK(U11(isNat(z0), z1)), U11'(isNat(z0), z1), ISNAT(z0))
ACTIVE(isNat(s(z0))) → c13(MARK(U21(isNat(z0))), U21'(isNat(z0)), ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c14(MARK(U31(isNat(z0), z1)), U31'(isNat(z0), z1), ISNAT(z0))
ACTIVE(plus(z0, 0)) → c15(MARK(U41(isNat(z0), z0)), U41'(isNat(z0), z0), ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c16(MARK(U51(isNat(z1), z1, z0)), U51'(isNat(z1), z1, z0), ISNAT(z1))
ACTIVE(x(z0, 0)) → c17(MARK(U61(isNat(z0))), U61'(isNat(z0)), ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c18(MARK(U71(isNat(z1), z1, z0)), U71'(isNat(z1), z1, z0), ISNAT(z1))
MARK(U11(z0, z1)) → c19(ACTIVE(U11(mark(z0), z1)), U11'(mark(z0), z1), MARK(z0))
MARK(tt) → c20(ACTIVE(tt))
MARK(U12(z0)) → c21(ACTIVE(U12(mark(z0))), U12'(mark(z0)), MARK(z0))
MARK(isNat(z0)) → c22(ACTIVE(isNat(z0)), ISNAT(z0))
MARK(U21(z0)) → c23(ACTIVE(U21(mark(z0))), U21'(mark(z0)), MARK(z0))
MARK(U31(z0, z1)) → c24(ACTIVE(U31(mark(z0), z1)), U31'(mark(z0), z1), MARK(z0))
MARK(U32(z0)) → c25(ACTIVE(U32(mark(z0))), U32'(mark(z0)), MARK(z0))
MARK(U41(z0, z1)) → c26(ACTIVE(U41(mark(z0), z1)), U41'(mark(z0), z1), MARK(z0))
MARK(U51(z0, z1, z2)) → c27(ACTIVE(U51(mark(z0), z1, z2)), U51'(mark(z0), z1, z2), MARK(z0))
MARK(U52(z0, z1, z2)) → c28(ACTIVE(U52(mark(z0), z1, z2)), U52'(mark(z0), z1, z2), MARK(z0))
MARK(s(z0)) → c29(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(plus(z0, z1)) → c30(ACTIVE(plus(mark(z0), mark(z1))), PLUS(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(U61(z0)) → c31(ACTIVE(U61(mark(z0))), U61'(mark(z0)), MARK(z0))
MARK(0) → c32(ACTIVE(0))
MARK(U71(z0, z1, z2)) → c33(ACTIVE(U71(mark(z0), z1, z2)), U71'(mark(z0), z1, z2), MARK(z0))
MARK(U72(z0, z1, z2)) → c34(ACTIVE(U72(mark(z0), z1, z2)), U72'(mark(z0), z1, z2), MARK(z0))
MARK(x(z0, z1)) → c35(ACTIVE(x(mark(z0), mark(z1))), X(mark(z0), mark(z1)), MARK(z0), MARK(z1))
U11'(mark(z0), z1) → c36(U11'(z0, z1))
U11'(z0, mark(z1)) → c37(U11'(z0, z1))
U11'(active(z0), z1) → c38(U11'(z0, z1))
U11'(z0, active(z1)) → c39(U11'(z0, z1))
U12'(mark(z0)) → c40(U12'(z0))
U12'(active(z0)) → c41(U12'(z0))
ISNAT(mark(z0)) → c42(ISNAT(z0))
ISNAT(active(z0)) → c43(ISNAT(z0))
U21'(mark(z0)) → c44(U21'(z0))
U21'(active(z0)) → c45(U21'(z0))
U31'(mark(z0), z1) → c46(U31'(z0, z1))
U31'(z0, mark(z1)) → c47(U31'(z0, z1))
U31'(active(z0), z1) → c48(U31'(z0, z1))
U31'(z0, active(z1)) → c49(U31'(z0, z1))
U32'(mark(z0)) → c50(U32'(z0))
U32'(active(z0)) → c51(U32'(z0))
U41'(mark(z0), z1) → c52(U41'(z0, z1))
U41'(z0, mark(z1)) → c53(U41'(z0, z1))
U41'(active(z0), z1) → c54(U41'(z0, z1))
U41'(z0, active(z1)) → c55(U41'(z0, z1))
U51'(mark(z0), z1, z2) → c56(U51'(z0, z1, z2))
U51'(z0, mark(z1), z2) → c57(U51'(z0, z1, z2))
U51'(z0, z1, mark(z2)) → c58(U51'(z0, z1, z2))
U51'(active(z0), z1, z2) → c59(U51'(z0, z1, z2))
U51'(z0, active(z1), z2) → c60(U51'(z0, z1, z2))
U51'(z0, z1, active(z2)) → c61(U51'(z0, z1, z2))
U52'(mark(z0), z1, z2) → c62(U52'(z0, z1, z2))
U52'(z0, mark(z1), z2) → c63(U52'(z0, z1, z2))
U52'(z0, z1, mark(z2)) → c64(U52'(z0, z1, z2))
U52'(active(z0), z1, z2) → c65(U52'(z0, z1, z2))
U52'(z0, active(z1), z2) → c66(U52'(z0, z1, z2))
U52'(z0, z1, active(z2)) → c67(U52'(z0, z1, z2))
S(mark(z0)) → c68(S(z0))
S(active(z0)) → c69(S(z0))
PLUS(mark(z0), z1) → c70(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c71(PLUS(z0, z1))
PLUS(active(z0), z1) → c72(PLUS(z0, z1))
PLUS(z0, active(z1)) → c73(PLUS(z0, z1))
U61'(mark(z0)) → c74(U61'(z0))
U61'(active(z0)) → c75(U61'(z0))
U71'(mark(z0), z1, z2) → c76(U71'(z0, z1, z2))
U71'(z0, mark(z1), z2) → c77(U71'(z0, z1, z2))
U71'(z0, z1, mark(z2)) → c78(U71'(z0, z1, z2))
U71'(active(z0), z1, z2) → c79(U71'(z0, z1, z2))
U71'(z0, active(z1), z2) → c80(U71'(z0, z1, z2))
U71'(z0, z1, active(z2)) → c81(U71'(z0, z1, z2))
U72'(mark(z0), z1, z2) → c82(U72'(z0, z1, z2))
U72'(z0, mark(z1), z2) → c83(U72'(z0, z1, z2))
U72'(z0, z1, mark(z2)) → c84(U72'(z0, z1, z2))
U72'(active(z0), z1, z2) → c85(U72'(z0, z1, z2))
U72'(z0, active(z1), z2) → c86(U72'(z0, z1, z2))
U72'(z0, z1, active(z2)) → c87(U72'(z0, z1, z2))
X(mark(z0), z1) → c88(X(z0, z1))
X(z0, mark(z1)) → c89(X(z0, z1))
X(active(z0), z1) → c90(X(z0, z1))
X(z0, active(z1)) → c91(X(z0, z1))
K tuples:none
Defined Rule Symbols:
active, mark, U11, U12, isNat, U21, U31, U32, U41, U51, U52, s, plus, U61, U71, U72, x
Defined Pair Symbols:
ACTIVE, MARK, U11', U12', ISNAT, U21', U31', U32', U41', U51', U52', S, PLUS, U61', U71', U72', X
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
(3) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
ACTIVE(U11(tt, z0)) → c(MARK(U12(isNat(z0))), U12'(isNat(z0)), ISNAT(z0))
ACTIVE(U12(tt)) → c1(MARK(tt))
ACTIVE(U21(tt)) → c2(MARK(tt))
ACTIVE(U31(tt, z0)) → c3(MARK(U32(isNat(z0))), U32'(isNat(z0)), ISNAT(z0))
ACTIVE(U32(tt)) → c4(MARK(tt))
ACTIVE(U41(tt, z0)) → c5(MARK(z0))
ACTIVE(U51(tt, z0, z1)) → c6(MARK(U52(isNat(z1), z0, z1)), U52'(isNat(z1), z0, z1), ISNAT(z1))
ACTIVE(U52(tt, z0, z1)) → c7(MARK(s(plus(z1, z0))), S(plus(z1, z0)), PLUS(z1, z0))
ACTIVE(U61(tt)) → c8(MARK(0))
ACTIVE(U71(tt, z0, z1)) → c9(MARK(U72(isNat(z1), z0, z1)), U72'(isNat(z1), z0, z1), ISNAT(z1))
ACTIVE(U72(tt, z0, z1)) → c10(MARK(plus(x(z1, z0), z1)), PLUS(x(z1, z0), z1), X(z1, z0))
ACTIVE(isNat(0)) → c11(MARK(tt))
ACTIVE(isNat(plus(z0, z1))) → c12(MARK(U11(isNat(z0), z1)), U11'(isNat(z0), z1), ISNAT(z0))
ACTIVE(isNat(s(z0))) → c13(MARK(U21(isNat(z0))), U21'(isNat(z0)), ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c14(MARK(U31(isNat(z0), z1)), U31'(isNat(z0), z1), ISNAT(z0))
ACTIVE(plus(z0, 0)) → c15(MARK(U41(isNat(z0), z0)), U41'(isNat(z0), z0), ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c16(MARK(U51(isNat(z1), z1, z0)), U51'(isNat(z1), z1, z0), ISNAT(z1))
ACTIVE(x(z0, 0)) → c17(MARK(U61(isNat(z0))), U61'(isNat(z0)), ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c18(MARK(U71(isNat(z1), z1, z0)), U71'(isNat(z1), z1, z0), ISNAT(z1))
MARK(U11(z0, z1)) → c19(ACTIVE(U11(mark(z0), z1)), U11'(mark(z0), z1), MARK(z0))
MARK(U12(z0)) → c21(ACTIVE(U12(mark(z0))), U12'(mark(z0)), MARK(z0))
MARK(isNat(z0)) → c22(ACTIVE(isNat(z0)), ISNAT(z0))
MARK(U21(z0)) → c23(ACTIVE(U21(mark(z0))), U21'(mark(z0)), MARK(z0))
MARK(U31(z0, z1)) → c24(ACTIVE(U31(mark(z0), z1)), U31'(mark(z0), z1), MARK(z0))
MARK(U32(z0)) → c25(ACTIVE(U32(mark(z0))), U32'(mark(z0)), MARK(z0))
MARK(U41(z0, z1)) → c26(ACTIVE(U41(mark(z0), z1)), U41'(mark(z0), z1), MARK(z0))
MARK(U51(z0, z1, z2)) → c27(ACTIVE(U51(mark(z0), z1, z2)), U51'(mark(z0), z1, z2), MARK(z0))
MARK(U52(z0, z1, z2)) → c28(ACTIVE(U52(mark(z0), z1, z2)), U52'(mark(z0), z1, z2), MARK(z0))
MARK(s(z0)) → c29(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(plus(z0, z1)) → c30(ACTIVE(plus(mark(z0), mark(z1))), PLUS(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(U61(z0)) → c31(ACTIVE(U61(mark(z0))), U61'(mark(z0)), MARK(z0))
MARK(U71(z0, z1, z2)) → c33(ACTIVE(U71(mark(z0), z1, z2)), U71'(mark(z0), z1, z2), MARK(z0))
MARK(U72(z0, z1, z2)) → c34(ACTIVE(U72(mark(z0), z1, z2)), U72'(mark(z0), z1, z2), MARK(z0))
MARK(x(z0, z1)) → c35(ACTIVE(x(mark(z0), mark(z1))), X(mark(z0), mark(z1)), MARK(z0), MARK(z1))
U11'(mark(z0), z1) → c36(U11'(z0, z1))
U11'(z0, mark(z1)) → c37(U11'(z0, z1))
U11'(active(z0), z1) → c38(U11'(z0, z1))
U11'(z0, active(z1)) → c39(U11'(z0, z1))
U12'(mark(z0)) → c40(U12'(z0))
U12'(active(z0)) → c41(U12'(z0))
ISNAT(mark(z0)) → c42(ISNAT(z0))
ISNAT(active(z0)) → c43(ISNAT(z0))
U21'(mark(z0)) → c44(U21'(z0))
U21'(active(z0)) → c45(U21'(z0))
U31'(mark(z0), z1) → c46(U31'(z0, z1))
U31'(z0, mark(z1)) → c47(U31'(z0, z1))
U31'(active(z0), z1) → c48(U31'(z0, z1))
U31'(z0, active(z1)) → c49(U31'(z0, z1))
U32'(mark(z0)) → c50(U32'(z0))
U32'(active(z0)) → c51(U32'(z0))
U41'(mark(z0), z1) → c52(U41'(z0, z1))
U41'(z0, mark(z1)) → c53(U41'(z0, z1))
U41'(active(z0), z1) → c54(U41'(z0, z1))
U41'(z0, active(z1)) → c55(U41'(z0, z1))
U51'(mark(z0), z1, z2) → c56(U51'(z0, z1, z2))
U51'(z0, mark(z1), z2) → c57(U51'(z0, z1, z2))
U51'(z0, z1, mark(z2)) → c58(U51'(z0, z1, z2))
U51'(active(z0), z1, z2) → c59(U51'(z0, z1, z2))
U51'(z0, active(z1), z2) → c60(U51'(z0, z1, z2))
U51'(z0, z1, active(z2)) → c61(U51'(z0, z1, z2))
U52'(mark(z0), z1, z2) → c62(U52'(z0, z1, z2))
U52'(z0, mark(z1), z2) → c63(U52'(z0, z1, z2))
U52'(z0, z1, mark(z2)) → c64(U52'(z0, z1, z2))
U52'(active(z0), z1, z2) → c65(U52'(z0, z1, z2))
U52'(z0, active(z1), z2) → c66(U52'(z0, z1, z2))
U52'(z0, z1, active(z2)) → c67(U52'(z0, z1, z2))
S(mark(z0)) → c68(S(z0))
S(active(z0)) → c69(S(z0))
PLUS(mark(z0), z1) → c70(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c71(PLUS(z0, z1))
PLUS(active(z0), z1) → c72(PLUS(z0, z1))
PLUS(z0, active(z1)) → c73(PLUS(z0, z1))
U61'(mark(z0)) → c74(U61'(z0))
U61'(active(z0)) → c75(U61'(z0))
U71'(mark(z0), z1, z2) → c76(U71'(z0, z1, z2))
U71'(z0, mark(z1), z2) → c77(U71'(z0, z1, z2))
U71'(z0, z1, mark(z2)) → c78(U71'(z0, z1, z2))
U71'(active(z0), z1, z2) → c79(U71'(z0, z1, z2))
U71'(z0, active(z1), z2) → c80(U71'(z0, z1, z2))
U71'(z0, z1, active(z2)) → c81(U71'(z0, z1, z2))
U72'(mark(z0), z1, z2) → c82(U72'(z0, z1, z2))
U72'(z0, mark(z1), z2) → c83(U72'(z0, z1, z2))
U72'(z0, z1, mark(z2)) → c84(U72'(z0, z1, z2))
U72'(active(z0), z1, z2) → c85(U72'(z0, z1, z2))
U72'(z0, active(z1), z2) → c86(U72'(z0, z1, z2))
U72'(z0, z1, active(z2)) → c87(U72'(z0, z1, z2))
X(mark(z0), z1) → c88(X(z0, z1))
X(z0, mark(z1)) → c89(X(z0, z1))
X(active(z0), z1) → c90(X(z0, z1))
X(z0, active(z1)) → c91(X(z0, z1))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0)) → mark(U12(isNat(z0)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, z0)) → mark(U32(isNat(z0)))
active(U32(tt)) → mark(tt)
active(U41(tt, z0)) → mark(z0)
active(U51(tt, z0, z1)) → mark(U52(isNat(z1), z0, z1))
active(U52(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U61(tt)) → mark(0)
active(U71(tt, z0, z1)) → mark(U72(isNat(z1), z0, z1))
active(U72(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNat(z0), z1))
active(isNat(s(z0))) → mark(U21(isNat(z0)))
active(isNat(x(z0, z1))) → mark(U31(isNat(z0), z1))
active(plus(z0, 0)) → mark(U41(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U51(isNat(z1), z1, z0))
active(x(z0, 0)) → mark(U61(isNat(z0)))
active(x(z0, s(z1))) → mark(U71(isNat(z1), z1, z0))
mark(U11(z0, z1)) → active(U11(mark(z0), z1))
mark(tt) → active(tt)
mark(U12(z0)) → active(U12(mark(z0)))
mark(isNat(z0)) → active(isNat(z0))
mark(U21(z0)) → active(U21(mark(z0)))
mark(U31(z0, z1)) → active(U31(mark(z0), z1))
mark(U32(z0)) → active(U32(mark(z0)))
mark(U41(z0, z1)) → active(U41(mark(z0), z1))
mark(U51(z0, z1, z2)) → active(U51(mark(z0), z1, z2))
mark(U52(z0, z1, z2)) → active(U52(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(U72(z0, z1, z2)) → active(U72(mark(z0), z1, z2))
mark(x(z0, z1)) → active(x(mark(z0), mark(z1)))
U11(mark(z0), z1) → U11(z0, z1)
U11(z0, mark(z1)) → U11(z0, z1)
U11(active(z0), z1) → U11(z0, z1)
U11(z0, active(z1)) → U11(z0, z1)
U12(mark(z0)) → U12(z0)
U12(active(z0)) → U12(z0)
isNat(mark(z0)) → isNat(z0)
isNat(active(z0)) → isNat(z0)
U21(mark(z0)) → U21(z0)
U21(active(z0)) → U21(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)
U32(mark(z0)) → U32(z0)
U32(active(z0)) → U32(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)
U52(mark(z0), z1, z2) → U52(z0, z1, z2)
U52(z0, mark(z1), z2) → U52(z0, z1, z2)
U52(z0, z1, mark(z2)) → U52(z0, z1, z2)
U52(active(z0), z1, z2) → U52(z0, z1, z2)
U52(z0, active(z1), z2) → U52(z0, z1, z2)
U52(z0, z1, active(z2)) → U52(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)
U72(mark(z0), z1, z2) → U72(z0, z1, z2)
U72(z0, mark(z1), z2) → U72(z0, z1, z2)
U72(z0, z1, mark(z2)) → U72(z0, z1, z2)
U72(active(z0), z1, z2) → U72(z0, z1, z2)
U72(z0, active(z1), z2) → U72(z0, z1, z2)
U72(z0, z1, active(z2)) → U72(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)
Tuples:
MARK(tt) → c20(ACTIVE(tt))
MARK(0) → c32(ACTIVE(0))
S tuples:
MARK(tt) → c20(ACTIVE(tt))
MARK(0) → c32(ACTIVE(0))
K tuples:none
Defined Rule Symbols:
active, mark, U11, U12, isNat, U21, U31, U32, U41, U51, U52, s, plus, U61, U71, U72, x
Defined Pair Symbols:
MARK
Compound Symbols:
c20, c32
(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing nodes:
MARK(0) → c32(ACTIVE(0))
MARK(tt) → c20(ACTIVE(tt))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0)) → mark(U12(isNat(z0)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, z0)) → mark(U32(isNat(z0)))
active(U32(tt)) → mark(tt)
active(U41(tt, z0)) → mark(z0)
active(U51(tt, z0, z1)) → mark(U52(isNat(z1), z0, z1))
active(U52(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U61(tt)) → mark(0)
active(U71(tt, z0, z1)) → mark(U72(isNat(z1), z0, z1))
active(U72(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNat(z0), z1))
active(isNat(s(z0))) → mark(U21(isNat(z0)))
active(isNat(x(z0, z1))) → mark(U31(isNat(z0), z1))
active(plus(z0, 0)) → mark(U41(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U51(isNat(z1), z1, z0))
active(x(z0, 0)) → mark(U61(isNat(z0)))
active(x(z0, s(z1))) → mark(U71(isNat(z1), z1, z0))
mark(U11(z0, z1)) → active(U11(mark(z0), z1))
mark(tt) → active(tt)
mark(U12(z0)) → active(U12(mark(z0)))
mark(isNat(z0)) → active(isNat(z0))
mark(U21(z0)) → active(U21(mark(z0)))
mark(U31(z0, z1)) → active(U31(mark(z0), z1))
mark(U32(z0)) → active(U32(mark(z0)))
mark(U41(z0, z1)) → active(U41(mark(z0), z1))
mark(U51(z0, z1, z2)) → active(U51(mark(z0), z1, z2))
mark(U52(z0, z1, z2)) → active(U52(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(U72(z0, z1, z2)) → active(U72(mark(z0), z1, z2))
mark(x(z0, z1)) → active(x(mark(z0), mark(z1)))
U11(mark(z0), z1) → U11(z0, z1)
U11(z0, mark(z1)) → U11(z0, z1)
U11(active(z0), z1) → U11(z0, z1)
U11(z0, active(z1)) → U11(z0, z1)
U12(mark(z0)) → U12(z0)
U12(active(z0)) → U12(z0)
isNat(mark(z0)) → isNat(z0)
isNat(active(z0)) → isNat(z0)
U21(mark(z0)) → U21(z0)
U21(active(z0)) → U21(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)
U32(mark(z0)) → U32(z0)
U32(active(z0)) → U32(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)
U52(mark(z0), z1, z2) → U52(z0, z1, z2)
U52(z0, mark(z1), z2) → U52(z0, z1, z2)
U52(z0, z1, mark(z2)) → U52(z0, z1, z2)
U52(active(z0), z1, z2) → U52(z0, z1, z2)
U52(z0, active(z1), z2) → U52(z0, z1, z2)
U52(z0, z1, active(z2)) → U52(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)
U72(mark(z0), z1, z2) → U72(z0, z1, z2)
U72(z0, mark(z1), z2) → U72(z0, z1, z2)
U72(z0, z1, mark(z2)) → U72(z0, z1, z2)
U72(active(z0), z1, z2) → U72(z0, z1, z2)
U72(z0, active(z1), z2) → U72(z0, z1, z2)
U72(z0, z1, active(z2)) → U72(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)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
active, mark, U11, U12, isNat, U21, U31, U32, U41, U51, U52, s, plus, U61, U71, U72, x
Defined Pair Symbols:none
Compound Symbols:none
(7) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(8) BOUNDS(O(1), O(1))