(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(and(tt, X)) → mark(X)
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(s(plus(N, M)))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(plus(x(N, M), N))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(tt) → active(tt)
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(0) → active(0)
mark(s(X)) → active(s(mark(X)))
mark(x(X1, X2)) → active(x(mark(X1), mark(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)
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)
s(mark(X)) → s(X)
s(active(X)) → s(X)
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(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(x(z0, 0)) → mark(0)
active(x(z0, s(z1))) → mark(plus(x(z0, z1), z0))
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(plus(z0, z1)) → active(plus(mark(z0), mark(z1)))
mark(0) → active(0)
mark(s(z0)) → active(s(mark(z0)))
mark(x(z0, z1)) → active(x(mark(z0), mark(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)
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)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
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(and(tt, z0)) → c(MARK(z0))
ACTIVE(plus(z0, 0)) → c1(MARK(z0))
ACTIVE(plus(z0, s(z1))) → c2(MARK(s(plus(z0, z1))), S(plus(z0, z1)), PLUS(z0, z1))
ACTIVE(x(z0, 0)) → c3(MARK(0))
ACTIVE(x(z0, s(z1))) → c4(MARK(plus(x(z0, z1), z0)), PLUS(x(z0, z1), z0), X(z0, z1))
MARK(and(z0, z1)) → c5(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(tt) → c6(ACTIVE(tt))
MARK(plus(z0, z1)) → c7(ACTIVE(plus(mark(z0), mark(z1))), PLUS(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(0) → c8(ACTIVE(0))
MARK(s(z0)) → c9(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(x(z0, z1)) → c10(ACTIVE(x(mark(z0), mark(z1))), X(mark(z0), mark(z1)), MARK(z0), MARK(z1))
AND(mark(z0), z1) → c11(AND(z0, z1))
AND(z0, mark(z1)) → c12(AND(z0, z1))
AND(active(z0), z1) → c13(AND(z0, z1))
AND(z0, active(z1)) → c14(AND(z0, z1))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c16(PLUS(z0, z1))
PLUS(active(z0), z1) → c17(PLUS(z0, z1))
PLUS(z0, active(z1)) → c18(PLUS(z0, z1))
S(mark(z0)) → c19(S(z0))
S(active(z0)) → c20(S(z0))
X(mark(z0), z1) → c21(X(z0, z1))
X(z0, mark(z1)) → c22(X(z0, z1))
X(active(z0), z1) → c23(X(z0, z1))
X(z0, active(z1)) → c24(X(z0, z1))
S tuples:
ACTIVE(and(tt, z0)) → c(MARK(z0))
ACTIVE(plus(z0, 0)) → c1(MARK(z0))
ACTIVE(plus(z0, s(z1))) → c2(MARK(s(plus(z0, z1))), S(plus(z0, z1)), PLUS(z0, z1))
ACTIVE(x(z0, 0)) → c3(MARK(0))
ACTIVE(x(z0, s(z1))) → c4(MARK(plus(x(z0, z1), z0)), PLUS(x(z0, z1), z0), X(z0, z1))
MARK(and(z0, z1)) → c5(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(tt) → c6(ACTIVE(tt))
MARK(plus(z0, z1)) → c7(ACTIVE(plus(mark(z0), mark(z1))), PLUS(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(0) → c8(ACTIVE(0))
MARK(s(z0)) → c9(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(x(z0, z1)) → c10(ACTIVE(x(mark(z0), mark(z1))), X(mark(z0), mark(z1)), MARK(z0), MARK(z1))
AND(mark(z0), z1) → c11(AND(z0, z1))
AND(z0, mark(z1)) → c12(AND(z0, z1))
AND(active(z0), z1) → c13(AND(z0, z1))
AND(z0, active(z1)) → c14(AND(z0, z1))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c16(PLUS(z0, z1))
PLUS(active(z0), z1) → c17(PLUS(z0, z1))
PLUS(z0, active(z1)) → c18(PLUS(z0, z1))
S(mark(z0)) → c19(S(z0))
S(active(z0)) → c20(S(z0))
X(mark(z0), z1) → c21(X(z0, z1))
X(z0, mark(z1)) → c22(X(z0, z1))
X(active(z0), z1) → c23(X(z0, z1))
X(z0, active(z1)) → c24(X(z0, z1))
K tuples:none
Defined Rule Symbols:
active, mark, and, plus, s, x
Defined Pair Symbols:
ACTIVE, MARK, AND, PLUS, S, 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
(3) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
ACTIVE(and(tt, z0)) → c(MARK(z0))
ACTIVE(plus(z0, 0)) → c1(MARK(z0))
ACTIVE(plus(z0, s(z1))) → c2(MARK(s(plus(z0, z1))), S(plus(z0, z1)), PLUS(z0, z1))
ACTIVE(x(z0, 0)) → c3(MARK(0))
ACTIVE(x(z0, s(z1))) → c4(MARK(plus(x(z0, z1), z0)), PLUS(x(z0, z1), z0), X(z0, z1))
MARK(and(z0, z1)) → c5(ACTIVE(and(mark(z0), z1)), AND(mark(z0), z1), MARK(z0))
MARK(plus(z0, z1)) → c7(ACTIVE(plus(mark(z0), mark(z1))), PLUS(mark(z0), mark(z1)), MARK(z0), MARK(z1))
MARK(s(z0)) → c9(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(x(z0, z1)) → c10(ACTIVE(x(mark(z0), mark(z1))), X(mark(z0), mark(z1)), MARK(z0), MARK(z1))
AND(mark(z0), z1) → c11(AND(z0, z1))
AND(z0, mark(z1)) → c12(AND(z0, z1))
AND(active(z0), z1) → c13(AND(z0, z1))
AND(z0, active(z1)) → c14(AND(z0, z1))
PLUS(mark(z0), z1) → c15(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c16(PLUS(z0, z1))
PLUS(active(z0), z1) → c17(PLUS(z0, z1))
PLUS(z0, active(z1)) → c18(PLUS(z0, z1))
S(mark(z0)) → c19(S(z0))
S(active(z0)) → c20(S(z0))
X(mark(z0), z1) → c21(X(z0, z1))
X(z0, mark(z1)) → c22(X(z0, z1))
X(active(z0), z1) → c23(X(z0, z1))
X(z0, active(z1)) → c24(X(z0, z1))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(x(z0, 0)) → mark(0)
active(x(z0, s(z1))) → mark(plus(x(z0, z1), z0))
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(plus(z0, z1)) → active(plus(mark(z0), mark(z1)))
mark(0) → active(0)
mark(s(z0)) → active(s(mark(z0)))
mark(x(z0, z1)) → active(x(mark(z0), mark(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)
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)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
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) → c6(ACTIVE(tt))
MARK(0) → c8(ACTIVE(0))
S tuples:
MARK(tt) → c6(ACTIVE(tt))
MARK(0) → c8(ACTIVE(0))
K tuples:none
Defined Rule Symbols:
active, mark, and, plus, s, x
Defined Pair Symbols:
MARK
Compound Symbols:
c6, c8
(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing nodes:
MARK(tt) → c6(ACTIVE(tt))
MARK(0) → c8(ACTIVE(0))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(and(tt, z0)) → mark(z0)
active(plus(z0, 0)) → mark(z0)
active(plus(z0, s(z1))) → mark(s(plus(z0, z1)))
active(x(z0, 0)) → mark(0)
active(x(z0, s(z1))) → mark(plus(x(z0, z1), z0))
mark(and(z0, z1)) → active(and(mark(z0), z1))
mark(tt) → active(tt)
mark(plus(z0, z1)) → active(plus(mark(z0), mark(z1)))
mark(0) → active(0)
mark(s(z0)) → active(s(mark(z0)))
mark(x(z0, z1)) → active(x(mark(z0), mark(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)
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)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
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, and, plus, s, x
Defined Pair Symbols:none
Compound Symbols:none
(7) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(8) BOUNDS(O(1), O(1))