(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(f(0)) → mark(cons(0, f(s(0))))
active(f(s(0))) → mark(f(p(s(0))))
active(p(s(X))) → mark(X)
mark(f(X)) → active(f(mark(X)))
mark(0) → active(0)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(p(X)) → active(p(mark(X)))
f(mark(X)) → f(X)
f(active(X)) → f(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
p(mark(X)) → p(X)
p(active(X)) → p(X)
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(f(0)) → mark(cons(0, f(s(0))))
active(f(s(0))) → mark(f(p(s(0))))
active(p(s(z0))) → mark(z0)
mark(f(z0)) → active(f(mark(z0)))
mark(0) → active(0)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(s(z0)) → active(s(mark(z0)))
mark(p(z0)) → active(p(mark(z0)))
f(mark(z0)) → f(z0)
f(active(z0)) → f(z0)
cons(mark(z0), z1) → cons(z0, z1)
cons(z0, mark(z1)) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
p(mark(z0)) → p(z0)
p(active(z0)) → p(z0)
Tuples:
ACTIVE(f(0)) → c(MARK(cons(0, f(s(0)))), CONS(0, f(s(0))), F(s(0)), S(0))
ACTIVE(f(s(0))) → c1(MARK(f(p(s(0)))), F(p(s(0))), P(s(0)), S(0))
ACTIVE(p(s(z0))) → c2(MARK(z0))
MARK(f(z0)) → c3(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(0) → c4(ACTIVE(0))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(s(z0)) → c6(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(p(z0)) → c7(ACTIVE(p(mark(z0))), P(mark(z0)), MARK(z0))
F(mark(z0)) → c8(F(z0))
F(active(z0)) → c9(F(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(z0, mark(z1)) → c11(CONS(z0, z1))
CONS(active(z0), z1) → c12(CONS(z0, z1))
CONS(z0, active(z1)) → c13(CONS(z0, z1))
S(mark(z0)) → c14(S(z0))
S(active(z0)) → c15(S(z0))
P(mark(z0)) → c16(P(z0))
P(active(z0)) → c17(P(z0))
S tuples:
ACTIVE(f(0)) → c(MARK(cons(0, f(s(0)))), CONS(0, f(s(0))), F(s(0)), S(0))
ACTIVE(f(s(0))) → c1(MARK(f(p(s(0)))), F(p(s(0))), P(s(0)), S(0))
ACTIVE(p(s(z0))) → c2(MARK(z0))
MARK(f(z0)) → c3(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(0) → c4(ACTIVE(0))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(s(z0)) → c6(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(p(z0)) → c7(ACTIVE(p(mark(z0))), P(mark(z0)), MARK(z0))
F(mark(z0)) → c8(F(z0))
F(active(z0)) → c9(F(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(z0, mark(z1)) → c11(CONS(z0, z1))
CONS(active(z0), z1) → c12(CONS(z0, z1))
CONS(z0, active(z1)) → c13(CONS(z0, z1))
S(mark(z0)) → c14(S(z0))
S(active(z0)) → c15(S(z0))
P(mark(z0)) → c16(P(z0))
P(active(z0)) → c17(P(z0))
K tuples:none
Defined Rule Symbols:
active, mark, f, cons, s, p
Defined Pair Symbols:
ACTIVE, MARK, F, CONS, S, P
Compound Symbols:
c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17
(3) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
ACTIVE(f(0)) → c(MARK(cons(0, f(s(0)))), CONS(0, f(s(0))), F(s(0)), S(0))
ACTIVE(f(s(0))) → c1(MARK(f(p(s(0)))), F(p(s(0))), P(s(0)), S(0))
ACTIVE(p(s(z0))) → c2(MARK(z0))
MARK(f(z0)) → c3(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(cons(z0, z1)) → c5(ACTIVE(cons(mark(z0), z1)), CONS(mark(z0), z1), MARK(z0))
MARK(s(z0)) → c6(ACTIVE(s(mark(z0))), S(mark(z0)), MARK(z0))
MARK(p(z0)) → c7(ACTIVE(p(mark(z0))), P(mark(z0)), MARK(z0))
F(mark(z0)) → c8(F(z0))
F(active(z0)) → c9(F(z0))
CONS(mark(z0), z1) → c10(CONS(z0, z1))
CONS(z0, mark(z1)) → c11(CONS(z0, z1))
CONS(active(z0), z1) → c12(CONS(z0, z1))
CONS(z0, active(z1)) → c13(CONS(z0, z1))
S(mark(z0)) → c14(S(z0))
S(active(z0)) → c15(S(z0))
P(mark(z0)) → c16(P(z0))
P(active(z0)) → c17(P(z0))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(f(0)) → mark(cons(0, f(s(0))))
active(f(s(0))) → mark(f(p(s(0))))
active(p(s(z0))) → mark(z0)
mark(f(z0)) → active(f(mark(z0)))
mark(0) → active(0)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(s(z0)) → active(s(mark(z0)))
mark(p(z0)) → active(p(mark(z0)))
f(mark(z0)) → f(z0)
f(active(z0)) → f(z0)
cons(mark(z0), z1) → cons(z0, z1)
cons(z0, mark(z1)) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
p(mark(z0)) → p(z0)
p(active(z0)) → p(z0)
Tuples:
MARK(0) → c4(ACTIVE(0))
S tuples:
MARK(0) → c4(ACTIVE(0))
K tuples:none
Defined Rule Symbols:
active, mark, f, cons, s, p
Defined Pair Symbols:
MARK
Compound Symbols:
c4
(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
MARK(0) → c4(ACTIVE(0))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(f(0)) → mark(cons(0, f(s(0))))
active(f(s(0))) → mark(f(p(s(0))))
active(p(s(z0))) → mark(z0)
mark(f(z0)) → active(f(mark(z0)))
mark(0) → active(0)
mark(cons(z0, z1)) → active(cons(mark(z0), z1))
mark(s(z0)) → active(s(mark(z0)))
mark(p(z0)) → active(p(mark(z0)))
f(mark(z0)) → f(z0)
f(active(z0)) → f(z0)
cons(mark(z0), z1) → cons(z0, z1)
cons(z0, mark(z1)) → cons(z0, z1)
cons(active(z0), z1) → cons(z0, z1)
cons(z0, active(z1)) → cons(z0, z1)
s(mark(z0)) → s(z0)
s(active(z0)) → s(z0)
p(mark(z0)) → p(z0)
p(active(z0)) → p(z0)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
active, mark, f, cons, s, p
Defined Pair Symbols:none
Compound Symbols:none
(7) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(8) BOUNDS(O(1), O(1))