(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
mark(U11(X1, X2, X3)) → active(U11(mark(X1), X2, X3))
mark(tt) → active(tt)
mark(U12(X1, X2, X3)) → active(U12(mark(X1), X2, X3))
mark(s(X)) → active(s(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(U21(X1, X2, X3)) → active(U21(mark(X1), X2, X3))
mark(U22(X1, X2, X3)) → active(U22(mark(X1), X2, X3))
mark(x(X1, X2)) → active(x(mark(X1), mark(X2)))
mark(0) → active(0)
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, X3) → U12(X1, X2, X3)
U12(X1, mark(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, mark(X3)) → U12(X1, X2, X3)
U12(active(X1), X2, X3) → U12(X1, X2, X3)
U12(X1, active(X2), X3) → U12(X1, X2, X3)
U12(X1, X2, active(X3)) → U12(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)
U21(mark(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, mark(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, mark(X3)) → U21(X1, X2, X3)
U21(active(X1), X2, X3) → U21(X1, X2, X3)
U21(X1, active(X2), X3) → U21(X1, X2, X3)
U21(X1, X2, active(X3)) → U21(X1, X2, X3)
U22(mark(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, mark(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, mark(X3)) → U22(X1, X2, X3)
U22(active(X1), X2, X3) → U22(X1, X2, X3)
U22(X1, active(X2), X3) → U22(X1, X2, X3)
U22(X1, X2, active(X3)) → U22(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, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U21(tt, z0, z1)) → mark(U22(tt, z0, z1))
active(U22(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(x(z0, 0)) → mark(0)
active(x(z0, s(z1))) → mark(U21(tt, z1, z0))
mark(U11(z0, z1, z2)) → active(U11(mark(z0), z1, z2))
mark(tt) → active(tt)
mark(U12(z0, z1, z2)) → active(U12(mark(z0), z1, z2))
mark(s(z0)) → active(s(mark(z0)))
mark(plus(z0, z1)) → active(plus(mark(z0), mark(z1)))
mark(U21(z0, z1, z2)) → active(U21(mark(z0), z1, z2))
mark(U22(z0, z1, z2)) → active(U22(mark(z0), z1, z2))
mark(x(z0, z1)) → active(x(mark(z0), mark(z1)))
mark(0) → active(0)
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, z2) → U12(z0, z1, z2)
U12(z0, mark(z1), z2) → U12(z0, z1, z2)
U12(z0, z1, mark(z2)) → U12(z0, z1, z2)
U12(active(z0), z1, z2) → U12(z0, z1, z2)
U12(z0, active(z1), z2) → U12(z0, z1, z2)
U12(z0, z1, active(z2)) → U12(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)
U21(mark(z0), z1, z2) → U21(z0, z1, z2)
U21(z0, mark(z1), z2) → U21(z0, z1, z2)
U21(z0, z1, mark(z2)) → U21(z0, z1, z2)
U21(active(z0), z1, z2) → U21(z0, z1, z2)
U21(z0, active(z1), z2) → U21(z0, z1, z2)
U21(z0, z1, active(z2)) → U21(z0, z1, z2)
U22(mark(z0), z1, z2) → U22(z0, z1, z2)
U22(z0, mark(z1), z2) → U22(z0, z1, z2)
U22(z0, z1, mark(z2)) → U22(z0, z1, z2)
U22(active(z0), z1, z2) → U22(z0, z1, z2)
U22(z0, active(z1), z2) → U22(z0, z1, z2)
U22(z0, z1, active(z2)) → U22(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, z1)) → c(MARK(U12(tt, z0, z1)), U12'(tt, z0, z1))
ACTIVE(U12(tt, z0, z1)) → c1(MARK(s(plus(z1, z0))), S(plus(z1, z0)), PLUS(z1, z0))
ACTIVE(U21(tt, z0, z1)) → c2(MARK(U22(tt, z0, z1)), U22'(tt, z0, z1))
ACTIVE(U22(tt, z0, z1)) → c3(MARK(plus(x(z1, z0), z1)), PLUS(x(z1, z0), z1), X(z1, z0))
ACTIVE(plus(z0, 0)) → c4(MARK(z0))
ACTIVE(plus(z0, s(z1))) → c5(MARK(U11(tt, z1, z0)), U11'(tt, z1, z0))
ACTIVE(x(z0, 0)) → c6(MARK(0))
ACTIVE(x(z0, s(z1))) → c7(MARK(U21(tt, z1, z0)), U21'(tt, z1, z0))
MARK(U11(z0, z1, z2)) → c8(ACTIVE(U11(mark(z0), z1, z2)), U11'(mark(z0), z1, z2), MARK(z0))
MARK(tt) → c9(ACTIVE(tt))
MARK(U12(z0, z1, z2)) → c10(ACTIVE(U12(mark(z0), z1, z2)), U12'(mark(z0), z1, z2), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(plus(z0, z1)) → c12(ACTIVE(plus(mark(z0), mark(z1))), PLUS(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(U21(z0, z1, z2)) → c13(ACTIVE(U21(mark(z0), z1, z2)), U21'(mark(z0), z1, z2), MARK(z0))
MARK(U22(z0, z1, z2)) → c14(ACTIVE(U22(mark(z0), z1, z2)), U22'(mark(z0), z1, z2), MARK(z0))
MARK(x(z0, z1)) → c15(ACTIVE(x(mark(z0), mark(z1))), X(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(0) → c16(ACTIVE(0))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(z0, mark(z1), z2) → c18(U11'(z0, z1, z2))
U11'(z0, z1, mark(z2)) → c19(U11'(z0, z1, z2))
U11'(active(z0), z1, z2) → c20(U11'(z0, z1, z2))
U11'(z0, active(z1), z2) → c21(U11'(z0, z1, z2))
U11'(z0, z1, active(z2)) → c22(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c23(U12'(z0, z1, z2))
U12'(z0, mark(z1), z2) → c24(U12'(z0, z1, z2))
U12'(z0, z1, mark(z2)) → c25(U12'(z0, z1, z2))
U12'(active(z0), z1, z2) → c26(U12'(z0, z1, z2))
U12'(z0, active(z1), z2) → c27(U12'(z0, z1, z2))
U12'(z0, z1, active(z2)) → c28(U12'(z0, z1, z2))
S(mark(z0)) → c29(S(z0))
S(active(z0)) → c30(S(z0))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c32(PLUS(z0, z1))
PLUS(active(z0), z1) → c33(PLUS(z0, z1))
PLUS(z0, active(z1)) → c34(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c35(U21'(z0, z1, z2))
U21'(z0, mark(z1), z2) → c36(U21'(z0, z1, z2))
U21'(z0, z1, mark(z2)) → c37(U21'(z0, z1, z2))
U21'(active(z0), z1, z2) → c38(U21'(z0, z1, z2))
U21'(z0, active(z1), z2) → c39(U21'(z0, z1, z2))
U21'(z0, z1, active(z2)) → c40(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c41(U22'(z0, z1, z2))
U22'(z0, mark(z1), z2) → c42(U22'(z0, z1, z2))
U22'(z0, z1, mark(z2)) → c43(U22'(z0, z1, z2))
U22'(active(z0), z1, z2) → c44(U22'(z0, z1, z2))
U22'(z0, active(z1), z2) → c45(U22'(z0, z1, z2))
U22'(z0, z1, active(z2)) → c46(U22'(z0, z1, z2))
X(mark(z0), z1) → c47(X(z0, z1))
X(z0, mark(z1)) → c48(X(z0, z1))
X(active(z0), z1) → c49(X(z0, z1))
X(z0, active(z1)) → c50(X(z0, z1))
S tuples:
ACTIVE(U11(tt, z0, z1)) → c(MARK(U12(tt, z0, z1)), U12'(tt, z0, z1))
ACTIVE(U12(tt, z0, z1)) → c1(MARK(s(plus(z1, z0))), S(plus(z1, z0)), PLUS(z1, z0))
ACTIVE(U21(tt, z0, z1)) → c2(MARK(U22(tt, z0, z1)), U22'(tt, z0, z1))
ACTIVE(U22(tt, z0, z1)) → c3(MARK(plus(x(z1, z0), z1)), PLUS(x(z1, z0), z1), X(z1, z0))
ACTIVE(plus(z0, 0)) → c4(MARK(z0))
ACTIVE(plus(z0, s(z1))) → c5(MARK(U11(tt, z1, z0)), U11'(tt, z1, z0))
ACTIVE(x(z0, 0)) → c6(MARK(0))
ACTIVE(x(z0, s(z1))) → c7(MARK(U21(tt, z1, z0)), U21'(tt, z1, z0))
MARK(U11(z0, z1, z2)) → c8(ACTIVE(U11(mark(z0), z1, z2)), U11'(mark(z0), z1, z2), MARK(z0))
MARK(tt) → c9(ACTIVE(tt))
MARK(U12(z0, z1, z2)) → c10(ACTIVE(U12(mark(z0), z1, z2)), U12'(mark(z0), z1, z2), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(plus(z0, z1)) → c12(ACTIVE(plus(mark(z0), mark(z1))), PLUS(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(U21(z0, z1, z2)) → c13(ACTIVE(U21(mark(z0), z1, z2)), U21'(mark(z0), z1, z2), MARK(z0))
MARK(U22(z0, z1, z2)) → c14(ACTIVE(U22(mark(z0), z1, z2)), U22'(mark(z0), z1, z2), MARK(z0))
MARK(x(z0, z1)) → c15(ACTIVE(x(mark(z0), mark(z1))), X(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(0) → c16(ACTIVE(0))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(z0, mark(z1), z2) → c18(U11'(z0, z1, z2))
U11'(z0, z1, mark(z2)) → c19(U11'(z0, z1, z2))
U11'(active(z0), z1, z2) → c20(U11'(z0, z1, z2))
U11'(z0, active(z1), z2) → c21(U11'(z0, z1, z2))
U11'(z0, z1, active(z2)) → c22(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c23(U12'(z0, z1, z2))
U12'(z0, mark(z1), z2) → c24(U12'(z0, z1, z2))
U12'(z0, z1, mark(z2)) → c25(U12'(z0, z1, z2))
U12'(active(z0), z1, z2) → c26(U12'(z0, z1, z2))
U12'(z0, active(z1), z2) → c27(U12'(z0, z1, z2))
U12'(z0, z1, active(z2)) → c28(U12'(z0, z1, z2))
S(mark(z0)) → c29(S(z0))
S(active(z0)) → c30(S(z0))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c32(PLUS(z0, z1))
PLUS(active(z0), z1) → c33(PLUS(z0, z1))
PLUS(z0, active(z1)) → c34(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c35(U21'(z0, z1, z2))
U21'(z0, mark(z1), z2) → c36(U21'(z0, z1, z2))
U21'(z0, z1, mark(z2)) → c37(U21'(z0, z1, z2))
U21'(active(z0), z1, z2) → c38(U21'(z0, z1, z2))
U21'(z0, active(z1), z2) → c39(U21'(z0, z1, z2))
U21'(z0, z1, active(z2)) → c40(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c41(U22'(z0, z1, z2))
U22'(z0, mark(z1), z2) → c42(U22'(z0, z1, z2))
U22'(z0, z1, mark(z2)) → c43(U22'(z0, z1, z2))
U22'(active(z0), z1, z2) → c44(U22'(z0, z1, z2))
U22'(z0, active(z1), z2) → c45(U22'(z0, z1, z2))
U22'(z0, z1, active(z2)) → c46(U22'(z0, z1, z2))
X(mark(z0), z1) → c47(X(z0, z1))
X(z0, mark(z1)) → c48(X(z0, z1))
X(active(z0), z1) → c49(X(z0, z1))
X(z0, active(z1)) → c50(X(z0, z1))
K tuples:none
Defined Rule Symbols:
active, mark, U11, U12, s, plus, U21, U22, x
Defined Pair Symbols:
ACTIVE, MARK, U11', U12', S, PLUS, U21', U22', 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
(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(tt, z0, z1)), U12'(tt, z0, z1))
ACTIVE(U12(tt, z0, z1)) → c1(MARK(s(plus(z1, z0))), S(plus(z1, z0)), PLUS(z1, z0))
ACTIVE(U21(tt, z0, z1)) → c2(MARK(U22(tt, z0, z1)), U22'(tt, z0, z1))
ACTIVE(U22(tt, z0, z1)) → c3(MARK(plus(x(z1, z0), z1)), PLUS(x(z1, z0), z1), X(z1, z0))
ACTIVE(plus(z0, 0)) → c4(MARK(z0))
ACTIVE(plus(z0, s(z1))) → c5(MARK(U11(tt, z1, z0)), U11'(tt, z1, z0))
ACTIVE(x(z0, 0)) → c6(MARK(0))
ACTIVE(x(z0, s(z1))) → c7(MARK(U21(tt, z1, z0)), U21'(tt, z1, z0))
MARK(U11(z0, z1, z2)) → c8(ACTIVE(U11(mark(z0), z1, z2)), U11'(mark(z0), z1, z2), MARK(z0))
MARK(U12(z0, z1, z2)) → c10(ACTIVE(U12(mark(z0), z1, z2)), U12'(mark(z0), z1, z2), MARK(z0))
MARK(s(z0)) → c11(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(plus(z0, z1)) → c12(ACTIVE(plus(mark(z0), mark(z1))), PLUS(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(U21(z0, z1, z2)) → c13(ACTIVE(U21(mark(z0), z1, z2)), U21'(mark(z0), z1, z2), MARK(z0))
MARK(U22(z0, z1, z2)) → c14(ACTIVE(U22(mark(z0), z1, z2)), U22'(mark(z0), z1, z2), MARK(z0))
MARK(x(z0, z1)) → c15(ACTIVE(x(mark(z0), mark(z1))), X(mark(z0), mark(z1)), MARK(z0), MARK(z1))
U11'(mark(z0), z1, z2) → c17(U11'(z0, z1, z2))
U11'(z0, mark(z1), z2) → c18(U11'(z0, z1, z2))
U11'(z0, z1, mark(z2)) → c19(U11'(z0, z1, z2))
U11'(active(z0), z1, z2) → c20(U11'(z0, z1, z2))
U11'(z0, active(z1), z2) → c21(U11'(z0, z1, z2))
U11'(z0, z1, active(z2)) → c22(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c23(U12'(z0, z1, z2))
U12'(z0, mark(z1), z2) → c24(U12'(z0, z1, z2))
U12'(z0, z1, mark(z2)) → c25(U12'(z0, z1, z2))
U12'(active(z0), z1, z2) → c26(U12'(z0, z1, z2))
U12'(z0, active(z1), z2) → c27(U12'(z0, z1, z2))
U12'(z0, z1, active(z2)) → c28(U12'(z0, z1, z2))
S(mark(z0)) → c29(S(z0))
S(active(z0)) → c30(S(z0))
PLUS(mark(z0), z1) → c31(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c32(PLUS(z0, z1))
PLUS(active(z0), z1) → c33(PLUS(z0, z1))
PLUS(z0, active(z1)) → c34(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c35(U21'(z0, z1, z2))
U21'(z0, mark(z1), z2) → c36(U21'(z0, z1, z2))
U21'(z0, z1, mark(z2)) → c37(U21'(z0, z1, z2))
U21'(active(z0), z1, z2) → c38(U21'(z0, z1, z2))
U21'(z0, active(z1), z2) → c39(U21'(z0, z1, z2))
U21'(z0, z1, active(z2)) → c40(U21'(z0, z1, z2))
U22'(mark(z0), z1, z2) → c41(U22'(z0, z1, z2))
U22'(z0, mark(z1), z2) → c42(U22'(z0, z1, z2))
U22'(z0, z1, mark(z2)) → c43(U22'(z0, z1, z2))
U22'(active(z0), z1, z2) → c44(U22'(z0, z1, z2))
U22'(z0, active(z1), z2) → c45(U22'(z0, z1, z2))
U22'(z0, z1, active(z2)) → c46(U22'(z0, z1, z2))
X(mark(z0), z1) → c47(X(z0, z1))
X(z0, mark(z1)) → c48(X(z0, z1))
X(active(z0), z1) → c49(X(z0, z1))
X(z0, active(z1)) → c50(X(z0, z1))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U21(tt, z0, z1)) → mark(U22(tt, z0, z1))
active(U22(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(x(z0, 0)) → mark(0)
active(x(z0, s(z1))) → mark(U21(tt, z1, z0))
mark(U11(z0, z1, z2)) → active(U11(mark(z0), z1, z2))
mark(tt) → active(tt)
mark(U12(z0, z1, z2)) → active(U12(mark(z0), z1, z2))
mark(s(z0)) → active(s(mark(z0)))
mark(plus(z0, z1)) → active(plus(mark(z0), mark(z1)))
mark(U21(z0, z1, z2)) → active(U21(mark(z0), z1, z2))
mark(U22(z0, z1, z2)) → active(U22(mark(z0), z1, z2))
mark(x(z0, z1)) → active(x(mark(z0), mark(z1)))
mark(0) → active(0)
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, z2) → U12(z0, z1, z2)
U12(z0, mark(z1), z2) → U12(z0, z1, z2)
U12(z0, z1, mark(z2)) → U12(z0, z1, z2)
U12(active(z0), z1, z2) → U12(z0, z1, z2)
U12(z0, active(z1), z2) → U12(z0, z1, z2)
U12(z0, z1, active(z2)) → U12(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)
U21(mark(z0), z1, z2) → U21(z0, z1, z2)
U21(z0, mark(z1), z2) → U21(z0, z1, z2)
U21(z0, z1, mark(z2)) → U21(z0, z1, z2)
U21(active(z0), z1, z2) → U21(z0, z1, z2)
U21(z0, active(z1), z2) → U21(z0, z1, z2)
U21(z0, z1, active(z2)) → U21(z0, z1, z2)
U22(mark(z0), z1, z2) → U22(z0, z1, z2)
U22(z0, mark(z1), z2) → U22(z0, z1, z2)
U22(z0, z1, mark(z2)) → U22(z0, z1, z2)
U22(active(z0), z1, z2) → U22(z0, z1, z2)
U22(z0, active(z1), z2) → U22(z0, z1, z2)
U22(z0, z1, active(z2)) → U22(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) → c9(ACTIVE(tt))
MARK(0) → c16(ACTIVE(0))
S tuples:
MARK(tt) → c9(ACTIVE(tt))
MARK(0) → c16(ACTIVE(0))
K tuples:none
Defined Rule Symbols:
active, mark, U11, U12, s, plus, U21, U22, x
Defined Pair Symbols:
MARK
Compound Symbols:
c9, c16
(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing nodes:
MARK(0) → c16(ACTIVE(0))
MARK(tt) → c9(ACTIVE(tt))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(U11(tt, z0, z1)) → mark(U12(tt, z0, z1))
active(U12(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U21(tt, z0, z1)) → mark(U22(tt, z0, z1))
active(U22(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(tt, z1, z0))
active(x(z0, 0)) → mark(0)
active(x(z0, s(z1))) → mark(U21(tt, z1, z0))
mark(U11(z0, z1, z2)) → active(U11(mark(z0), z1, z2))
mark(tt) → active(tt)
mark(U12(z0, z1, z2)) → active(U12(mark(z0), z1, z2))
mark(s(z0)) → active(s(mark(z0)))
mark(plus(z0, z1)) → active(plus(mark(z0), mark(z1)))
mark(U21(z0, z1, z2)) → active(U21(mark(z0), z1, z2))
mark(U22(z0, z1, z2)) → active(U22(mark(z0), z1, z2))
mark(x(z0, z1)) → active(x(mark(z0), mark(z1)))
mark(0) → active(0)
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, z2) → U12(z0, z1, z2)
U12(z0, mark(z1), z2) → U12(z0, z1, z2)
U12(z0, z1, mark(z2)) → U12(z0, z1, z2)
U12(active(z0), z1, z2) → U12(z0, z1, z2)
U12(z0, active(z1), z2) → U12(z0, z1, z2)
U12(z0, z1, active(z2)) → U12(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)
U21(mark(z0), z1, z2) → U21(z0, z1, z2)
U21(z0, mark(z1), z2) → U21(z0, z1, z2)
U21(z0, z1, mark(z2)) → U21(z0, z1, z2)
U21(active(z0), z1, z2) → U21(z0, z1, z2)
U21(z0, active(z1), z2) → U21(z0, z1, z2)
U21(z0, z1, active(z2)) → U21(z0, z1, z2)
U22(mark(z0), z1, z2) → U22(z0, z1, z2)
U22(z0, mark(z1), z2) → U22(z0, z1, z2)
U22(z0, z1, mark(z2)) → U22(z0, z1, z2)
U22(active(z0), z1, z2) → U22(z0, z1, z2)
U22(z0, active(z1), z2) → U22(z0, z1, z2)
U22(z0, z1, active(z2)) → U22(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, s, plus, U21, U22, x
Defined Pair Symbols:none
Compound Symbols:none
(7) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(8) BOUNDS(O(1), O(1))