(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(f(g(X), Y)) → mark(f(X, f(g(X), Y)))
mark(f(X1, X2)) → active(f(mark(X1), X2))
mark(g(X)) → active(g(mark(X)))
f(mark(X1), X2) → f(X1, X2)
f(X1, mark(X2)) → f(X1, X2)
f(active(X1), X2) → f(X1, X2)
f(X1, active(X2)) → f(X1, X2)
g(mark(X)) → g(X)
g(active(X)) → g(X)
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(f(g(z0), z1)) → mark(f(z0, f(g(z0), z1)))
mark(f(z0, z1)) → active(f(mark(z0), z1))
mark(g(z0)) → active(g(mark(z0)))
f(mark(z0), z1) → f(z0, z1)
f(z0, mark(z1)) → f(z0, z1)
f(active(z0), z1) → f(z0, z1)
f(z0, active(z1)) → f(z0, z1)
g(mark(z0)) → g(z0)
g(active(z0)) → g(z0)
Tuples:
ACTIVE(f(g(z0), z1)) → c(MARK(f(z0, f(g(z0), z1))), F(z0, f(g(z0), z1)), F(g(z0), z1), G(z0))
MARK(f(z0, z1)) → c1(ACTIVE(f(mark(z0), z1)), F(mark(z0), z1), MARK(z0))
MARK(g(z0)) → c2(ACTIVE(g(mark(z0))), G(mark(z0)), MARK(z0))
F(mark(z0), z1) → c3(F(z0, z1))
F(z0, mark(z1)) → c4(F(z0, z1))
F(active(z0), z1) → c5(F(z0, z1))
F(z0, active(z1)) → c6(F(z0, z1))
G(mark(z0)) → c7(G(z0))
G(active(z0)) → c8(G(z0))
S tuples:
ACTIVE(f(g(z0), z1)) → c(MARK(f(z0, f(g(z0), z1))), F(z0, f(g(z0), z1)), F(g(z0), z1), G(z0))
MARK(f(z0, z1)) → c1(ACTIVE(f(mark(z0), z1)), F(mark(z0), z1), MARK(z0))
MARK(g(z0)) → c2(ACTIVE(g(mark(z0))), G(mark(z0)), MARK(z0))
F(mark(z0), z1) → c3(F(z0, z1))
F(z0, mark(z1)) → c4(F(z0, z1))
F(active(z0), z1) → c5(F(z0, z1))
F(z0, active(z1)) → c6(F(z0, z1))
G(mark(z0)) → c7(G(z0))
G(active(z0)) → c8(G(z0))
K tuples:none
Defined Rule Symbols:
active, mark, f, g
Defined Pair Symbols:
ACTIVE, MARK, F, G
Compound Symbols:
c, c1, c2, c3, c4, c5, c6, c7, c8
(3) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
ACTIVE(f(g(z0), z1)) → c(MARK(f(z0, f(g(z0), z1))), F(z0, f(g(z0), z1)), F(g(z0), z1), G(z0))
MARK(f(z0, z1)) → c1(ACTIVE(f(mark(z0), z1)), F(mark(z0), z1), MARK(z0))
MARK(g(z0)) → c2(ACTIVE(g(mark(z0))), G(mark(z0)), MARK(z0))
F(mark(z0), z1) → c3(F(z0, z1))
F(z0, mark(z1)) → c4(F(z0, z1))
F(active(z0), z1) → c5(F(z0, z1))
F(z0, active(z1)) → c6(F(z0, z1))
G(mark(z0)) → c7(G(z0))
G(active(z0)) → c8(G(z0))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(f(g(z0), z1)) → mark(f(z0, f(g(z0), z1)))
mark(f(z0, z1)) → active(f(mark(z0), z1))
mark(g(z0)) → active(g(mark(z0)))
f(mark(z0), z1) → f(z0, z1)
f(z0, mark(z1)) → f(z0, z1)
f(active(z0), z1) → f(z0, z1)
f(z0, active(z1)) → f(z0, z1)
g(mark(z0)) → g(z0)
g(active(z0)) → g(z0)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
active, mark, f, g
Defined Pair Symbols:none
Compound Symbols:none
(5) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(6) BOUNDS(O(1), O(1))