(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(f(f(a))) → mark(f(g(f(a))))
mark(f(X)) → active(f(mark(X)))
mark(a) → active(a)
mark(g(X)) → active(g(X))
f(mark(X)) → f(X)
f(active(X)) → f(X)
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(f(a))) → mark(f(g(f(a))))
mark(f(z0)) → active(f(mark(z0)))
mark(a) → active(a)
mark(g(z0)) → active(g(z0))
f(mark(z0)) → f(z0)
f(active(z0)) → f(z0)
g(mark(z0)) → g(z0)
g(active(z0)) → g(z0)
Tuples:
ACTIVE(f(f(a))) → c(MARK(f(g(f(a)))), F(g(f(a))), G(f(a)), F(a))
MARK(f(z0)) → c1(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(a) → c2(ACTIVE(a))
MARK(g(z0)) → c3(ACTIVE(g(z0)), G(z0))
F(mark(z0)) → c4(F(z0))
F(active(z0)) → c5(F(z0))
G(mark(z0)) → c6(G(z0))
G(active(z0)) → c7(G(z0))
S tuples:
ACTIVE(f(f(a))) → c(MARK(f(g(f(a)))), F(g(f(a))), G(f(a)), F(a))
MARK(f(z0)) → c1(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(a) → c2(ACTIVE(a))
MARK(g(z0)) → c3(ACTIVE(g(z0)), G(z0))
F(mark(z0)) → c4(F(z0))
F(active(z0)) → c5(F(z0))
G(mark(z0)) → c6(G(z0))
G(active(z0)) → c7(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
(3) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
ACTIVE(f(f(a))) → c(MARK(f(g(f(a)))), F(g(f(a))), G(f(a)), F(a))
MARK(f(z0)) → c1(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(g(z0)) → c3(ACTIVE(g(z0)), G(z0))
F(mark(z0)) → c4(F(z0))
F(active(z0)) → c5(F(z0))
G(mark(z0)) → c6(G(z0))
G(active(z0)) → c7(G(z0))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(f(f(a))) → mark(f(g(f(a))))
mark(f(z0)) → active(f(mark(z0)))
mark(a) → active(a)
mark(g(z0)) → active(g(z0))
f(mark(z0)) → f(z0)
f(active(z0)) → f(z0)
g(mark(z0)) → g(z0)
g(active(z0)) → g(z0)
Tuples:
MARK(a) → c2(ACTIVE(a))
S tuples:
MARK(a) → c2(ACTIVE(a))
K tuples:none
Defined Rule Symbols:
active, mark, f, g
Defined Pair Symbols:
MARK
Compound Symbols:
c2
(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
MARK(a) → c2(ACTIVE(a))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(f(f(a))) → mark(f(g(f(a))))
mark(f(z0)) → active(f(mark(z0)))
mark(a) → active(a)
mark(g(z0)) → active(g(z0))
f(mark(z0)) → f(z0)
f(active(z0)) → f(z0)
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
(7) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(8) BOUNDS(O(1), O(1))