(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(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(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(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)
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(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(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(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)
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(plus(z0, 0)) → c2(MARK(z0))
ACTIVE(plus(z0, s(z1))) → c3(MARK(U11(tt, z1, z0)), U11'(tt, z1, z0))
MARK(U11(z0, z1, z2)) → c4(ACTIVE(U11(mark(z0), z1, z2)), U11'(mark(z0), z1, z2), MARK(z0))
MARK(tt) → c5(ACTIVE(tt))
MARK(U12(z0, z1, z2)) → c6(ACTIVE(U12(mark(z0), z1, z2)), U12'(mark(z0), z1, z2), MARK(z0))
MARK(s(z0)) → c7(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(plus(z0, z1)) → c8(ACTIVE(plus(mark(z0), mark(z1))), PLUS(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(0) → c9(ACTIVE(0))
U11'(mark(z0), z1, z2) → c10(U11'(z0, z1, z2))
U11'(z0, mark(z1), z2) → c11(U11'(z0, z1, z2))
U11'(z0, z1, mark(z2)) → c12(U11'(z0, z1, z2))
U11'(active(z0), z1, z2) → c13(U11'(z0, z1, z2))
U11'(z0, active(z1), z2) → c14(U11'(z0, z1, z2))
U11'(z0, z1, active(z2)) → c15(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c16(U12'(z0, z1, z2))
U12'(z0, mark(z1), z2) → c17(U12'(z0, z1, z2))
U12'(z0, z1, mark(z2)) → c18(U12'(z0, z1, z2))
U12'(active(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(z0, active(z1), z2) → c20(U12'(z0, z1, z2))
U12'(z0, z1, active(z2)) → c21(U12'(z0, z1, z2))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
PLUS(mark(z0), z1) → c24(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c25(PLUS(z0, z1))
PLUS(active(z0), z1) → c26(PLUS(z0, z1))
PLUS(z0, active(z1)) → c27(PLUS(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(plus(z0, 0)) → c2(MARK(z0))
ACTIVE(plus(z0, s(z1))) → c3(MARK(U11(tt, z1, z0)), U11'(tt, z1, z0))
MARK(U11(z0, z1, z2)) → c4(ACTIVE(U11(mark(z0), z1, z2)), U11'(mark(z0), z1, z2), MARK(z0))
MARK(tt) → c5(ACTIVE(tt))
MARK(U12(z0, z1, z2)) → c6(ACTIVE(U12(mark(z0), z1, z2)), U12'(mark(z0), z1, z2), MARK(z0))
MARK(s(z0)) → c7(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(plus(z0, z1)) → c8(ACTIVE(plus(mark(z0), mark(z1))), PLUS(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(0) → c9(ACTIVE(0))
U11'(mark(z0), z1, z2) → c10(U11'(z0, z1, z2))
U11'(z0, mark(z1), z2) → c11(U11'(z0, z1, z2))
U11'(z0, z1, mark(z2)) → c12(U11'(z0, z1, z2))
U11'(active(z0), z1, z2) → c13(U11'(z0, z1, z2))
U11'(z0, active(z1), z2) → c14(U11'(z0, z1, z2))
U11'(z0, z1, active(z2)) → c15(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c16(U12'(z0, z1, z2))
U12'(z0, mark(z1), z2) → c17(U12'(z0, z1, z2))
U12'(z0, z1, mark(z2)) → c18(U12'(z0, z1, z2))
U12'(active(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(z0, active(z1), z2) → c20(U12'(z0, z1, z2))
U12'(z0, z1, active(z2)) → c21(U12'(z0, z1, z2))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
PLUS(mark(z0), z1) → c24(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c25(PLUS(z0, z1))
PLUS(active(z0), z1) → c26(PLUS(z0, z1))
PLUS(z0, active(z1)) → c27(PLUS(z0, z1))
K tuples:none
Defined Rule Symbols:
active, mark, U11, U12, s, plus
Defined Pair Symbols:
ACTIVE, MARK, U11', U12', S, PLUS
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
(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(plus(z0, 0)) → c2(MARK(z0))
ACTIVE(plus(z0, s(z1))) → c3(MARK(U11(tt, z1, z0)), U11'(tt, z1, z0))
MARK(U11(z0, z1, z2)) → c4(ACTIVE(U11(mark(z0), z1, z2)), U11'(mark(z0), z1, z2), MARK(z0))
MARK(U12(z0, z1, z2)) → c6(ACTIVE(U12(mark(z0), z1, z2)), U12'(mark(z0), z1, z2), MARK(z0))
MARK(s(z0)) → c7(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(plus(z0, z1)) → c8(ACTIVE(plus(mark(z0), mark(z1))), PLUS(mark(z0), mark(z1)), MARK(z0), MARK(z1))
U11'(mark(z0), z1, z2) → c10(U11'(z0, z1, z2))
U11'(z0, mark(z1), z2) → c11(U11'(z0, z1, z2))
U11'(z0, z1, mark(z2)) → c12(U11'(z0, z1, z2))
U11'(active(z0), z1, z2) → c13(U11'(z0, z1, z2))
U11'(z0, active(z1), z2) → c14(U11'(z0, z1, z2))
U11'(z0, z1, active(z2)) → c15(U11'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c16(U12'(z0, z1, z2))
U12'(z0, mark(z1), z2) → c17(U12'(z0, z1, z2))
U12'(z0, z1, mark(z2)) → c18(U12'(z0, z1, z2))
U12'(active(z0), z1, z2) → c19(U12'(z0, z1, z2))
U12'(z0, active(z1), z2) → c20(U12'(z0, z1, z2))
U12'(z0, z1, active(z2)) → c21(U12'(z0, z1, z2))
S(mark(z0)) → c22(S(z0))
S(active(z0)) → c23(S(z0))
PLUS(mark(z0), z1) → c24(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c25(PLUS(z0, z1))
PLUS(active(z0), z1) → c26(PLUS(z0, z1))
PLUS(z0, active(z1)) → c27(PLUS(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(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(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(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)
Tuples:
MARK(tt) → c5(ACTIVE(tt))
MARK(0) → c9(ACTIVE(0))
S tuples:
MARK(tt) → c5(ACTIVE(tt))
MARK(0) → c9(ACTIVE(0))
K tuples:none
Defined Rule Symbols:
active, mark, U11, U12, s, plus
Defined Pair Symbols:
MARK
Compound Symbols:
c5, c9
(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing nodes:
MARK(0) → c9(ACTIVE(0))
MARK(tt) → c5(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(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(U11(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(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)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
active, mark, U11, U12, s, plus
Defined Pair Symbols:none
Compound Symbols:none
(7) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(8) BOUNDS(O(1), O(1))