(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(g(X)) → mark(h(X))
active(c) → mark(d)
active(h(d)) → mark(g(c))
mark(g(X)) → active(g(X))
mark(h(X)) → active(h(X))
mark(c) → active(c)
mark(d) → active(d)
g(mark(X)) → g(X)
g(active(X)) → g(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(g(z0)) → mark(h(z0))
active(c) → mark(d)
active(h(d)) → mark(g(c))
mark(g(z0)) → active(g(z0))
mark(h(z0)) → active(h(z0))
mark(c) → active(c)
mark(d) → active(d)
g(mark(z0)) → g(z0)
g(active(z0)) → g(z0)
h(mark(z0)) → h(z0)
h(active(z0)) → h(z0)
Tuples:
ACTIVE(g(z0)) → c1(MARK(h(z0)), H(z0))
ACTIVE(c) → c2(MARK(d))
ACTIVE(h(d)) → c3(MARK(g(c)), G(c))
MARK(g(z0)) → c4(ACTIVE(g(z0)), G(z0))
MARK(h(z0)) → c5(ACTIVE(h(z0)), H(z0))
MARK(c) → c6(ACTIVE(c))
MARK(d) → c7(ACTIVE(d))
G(mark(z0)) → c8(G(z0))
G(active(z0)) → c9(G(z0))
H(mark(z0)) → c10(H(z0))
H(active(z0)) → c11(H(z0))
S tuples:
ACTIVE(g(z0)) → c1(MARK(h(z0)), H(z0))
ACTIVE(c) → c2(MARK(d))
ACTIVE(h(d)) → c3(MARK(g(c)), G(c))
MARK(g(z0)) → c4(ACTIVE(g(z0)), G(z0))
MARK(h(z0)) → c5(ACTIVE(h(z0)), H(z0))
MARK(c) → c6(ACTIVE(c))
MARK(d) → c7(ACTIVE(d))
G(mark(z0)) → c8(G(z0))
G(active(z0)) → c9(G(z0))
H(mark(z0)) → c10(H(z0))
H(active(z0)) → c11(H(z0))
K tuples:none
Defined Rule Symbols:
active, mark, g, h
Defined Pair Symbols:
ACTIVE, MARK, G, H
Compound Symbols:
c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11
(3) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
ACTIVE(g(z0)) → c1(MARK(h(z0)), H(z0))
ACTIVE(h(d)) → c3(MARK(g(c)), G(c))
MARK(g(z0)) → c4(ACTIVE(g(z0)), G(z0))
MARK(h(z0)) → c5(ACTIVE(h(z0)), H(z0))
G(mark(z0)) → c8(G(z0))
G(active(z0)) → c9(G(z0))
H(mark(z0)) → c10(H(z0))
H(active(z0)) → c11(H(z0))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(g(z0)) → mark(h(z0))
active(c) → mark(d)
active(h(d)) → mark(g(c))
mark(g(z0)) → active(g(z0))
mark(h(z0)) → active(h(z0))
mark(c) → active(c)
mark(d) → active(d)
g(mark(z0)) → g(z0)
g(active(z0)) → g(z0)
h(mark(z0)) → h(z0)
h(active(z0)) → h(z0)
Tuples:
ACTIVE(c) → c2(MARK(d))
MARK(c) → c6(ACTIVE(c))
MARK(d) → c7(ACTIVE(d))
S tuples:
ACTIVE(c) → c2(MARK(d))
MARK(c) → c6(ACTIVE(c))
MARK(d) → c7(ACTIVE(d))
K tuples:none
Defined Rule Symbols:
active, mark, g, h
Defined Pair Symbols:
ACTIVE, MARK
Compound Symbols:
c2, c6, c7
(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 3 trailing nodes:
ACTIVE(c) → c2(MARK(d))
MARK(d) → c7(ACTIVE(d))
MARK(c) → c6(ACTIVE(c))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(g(z0)) → mark(h(z0))
active(c) → mark(d)
active(h(d)) → mark(g(c))
mark(g(z0)) → active(g(z0))
mark(h(z0)) → active(h(z0))
mark(c) → active(c)
mark(d) → active(d)
g(mark(z0)) → g(z0)
g(active(z0)) → g(z0)
h(mark(z0)) → h(z0)
h(active(z0)) → h(z0)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
active, mark, g, h
Defined Pair Symbols:none
Compound Symbols:none
(7) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(8) BOUNDS(O(1), O(1))