(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(f(f(X))) → mark(c(f(g(f(X)))))
active(c(X)) → mark(d(X))
active(h(X)) → mark(c(d(X)))
mark(f(X)) → active(f(mark(X)))
mark(c(X)) → active(c(X))
mark(g(X)) → active(g(X))
mark(d(X)) → active(d(X))
mark(h(X)) → active(h(mark(X)))
f(mark(X)) → f(X)
f(active(X)) → f(X)
c(mark(X)) → c(X)
c(active(X)) → c(X)
g(mark(X)) → g(X)
g(active(X)) → g(X)
d(mark(X)) → d(X)
d(active(X)) → d(X)
h(mark(X)) → h(X)
h(active(X)) → h(X)
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(f(f(z0))) → mark(c(f(g(f(z0)))))
active(c(z0)) → mark(d(z0))
active(h(z0)) → mark(c(d(z0)))
mark(f(z0)) → active(f(mark(z0)))
mark(c(z0)) → active(c(z0))
mark(g(z0)) → active(g(z0))
mark(d(z0)) → active(d(z0))
mark(h(z0)) → active(h(mark(z0)))
f(mark(z0)) → f(z0)
f(active(z0)) → f(z0)
c(mark(z0)) → c(z0)
c(active(z0)) → c(z0)
g(mark(z0)) → g(z0)
g(active(z0)) → g(z0)
d(mark(z0)) → d(z0)
d(active(z0)) → d(z0)
h(mark(z0)) → h(z0)
h(active(z0)) → h(z0)
Tuples:
ACTIVE(f(f(z0))) → c1(MARK(c(f(g(f(z0))))), C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(MARK(d(z0)), D(z0))
ACTIVE(h(z0)) → c3(MARK(c(d(z0))), C(d(z0)), D(z0))
MARK(f(z0)) → c4(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(c(z0)) → c5(ACTIVE(c(z0)), C(z0))
MARK(g(z0)) → c6(ACTIVE(g(z0)), G(z0))
MARK(d(z0)) → c7(ACTIVE(d(z0)), D(z0))
MARK(h(z0)) → c8(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
F(mark(z0)) → c9(F(z0))
F(active(z0)) → c10(F(z0))
C(mark(z0)) → c11(C(z0))
C(active(z0)) → c12(C(z0))
G(mark(z0)) → c13(G(z0))
G(active(z0)) → c14(G(z0))
D(mark(z0)) → c15(D(z0))
D(active(z0)) → c16(D(z0))
H(mark(z0)) → c17(H(z0))
H(active(z0)) → c18(H(z0))
S tuples:
ACTIVE(f(f(z0))) → c1(MARK(c(f(g(f(z0))))), C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(MARK(d(z0)), D(z0))
ACTIVE(h(z0)) → c3(MARK(c(d(z0))), C(d(z0)), D(z0))
MARK(f(z0)) → c4(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(c(z0)) → c5(ACTIVE(c(z0)), C(z0))
MARK(g(z0)) → c6(ACTIVE(g(z0)), G(z0))
MARK(d(z0)) → c7(ACTIVE(d(z0)), D(z0))
MARK(h(z0)) → c8(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
F(mark(z0)) → c9(F(z0))
F(active(z0)) → c10(F(z0))
C(mark(z0)) → c11(C(z0))
C(active(z0)) → c12(C(z0))
G(mark(z0)) → c13(G(z0))
G(active(z0)) → c14(G(z0))
D(mark(z0)) → c15(D(z0))
D(active(z0)) → c16(D(z0))
H(mark(z0)) → c17(H(z0))
H(active(z0)) → c18(H(z0))
K tuples:none
Defined Rule Symbols:
active, mark, f, c, g, d, h
Defined Pair Symbols:
ACTIVE, MARK, F, C, G, D, H
Compound Symbols:
c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18
(3) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
ACTIVE(f(f(z0))) → c1(MARK(c(f(g(f(z0))))), C(f(g(f(z0)))), F(g(f(z0))), G(f(z0)), F(z0))
ACTIVE(c(z0)) → c2(MARK(d(z0)), D(z0))
ACTIVE(h(z0)) → c3(MARK(c(d(z0))), C(d(z0)), D(z0))
MARK(f(z0)) → c4(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(c(z0)) → c5(ACTIVE(c(z0)), C(z0))
MARK(g(z0)) → c6(ACTIVE(g(z0)), G(z0))
MARK(d(z0)) → c7(ACTIVE(d(z0)), D(z0))
MARK(h(z0)) → c8(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
F(mark(z0)) → c9(F(z0))
F(active(z0)) → c10(F(z0))
C(mark(z0)) → c11(C(z0))
C(active(z0)) → c12(C(z0))
G(mark(z0)) → c13(G(z0))
G(active(z0)) → c14(G(z0))
D(mark(z0)) → c15(D(z0))
D(active(z0)) → c16(D(z0))
H(mark(z0)) → c17(H(z0))
H(active(z0)) → c18(H(z0))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(f(f(z0))) → mark(c(f(g(f(z0)))))
active(c(z0)) → mark(d(z0))
active(h(z0)) → mark(c(d(z0)))
mark(f(z0)) → active(f(mark(z0)))
mark(c(z0)) → active(c(z0))
mark(g(z0)) → active(g(z0))
mark(d(z0)) → active(d(z0))
mark(h(z0)) → active(h(mark(z0)))
f(mark(z0)) → f(z0)
f(active(z0)) → f(z0)
c(mark(z0)) → c(z0)
c(active(z0)) → c(z0)
g(mark(z0)) → g(z0)
g(active(z0)) → g(z0)
d(mark(z0)) → d(z0)
d(active(z0)) → d(z0)
h(mark(z0)) → h(z0)
h(active(z0)) → h(z0)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
active, mark, f, c, g, d, h
Defined Pair Symbols:none
Compound Symbols:none
(5) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(6) BOUNDS(O(1), O(1))