(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(f(X, g(X), Y)) → mark(f(Y, Y, Y))
active(g(b)) → mark(c)
active(b) → mark(c)
mark(f(X1, X2, X3)) → active(f(X1, X2, X3))
mark(g(X)) → active(g(mark(X)))
mark(b) → active(b)
mark(c) → active(c)
f(mark(X1), X2, X3) → f(X1, X2, X3)
f(X1, mark(X2), X3) → f(X1, X2, X3)
f(X1, X2, mark(X3)) → f(X1, X2, X3)
f(active(X1), X2, X3) → f(X1, X2, X3)
f(X1, active(X2), X3) → f(X1, X2, X3)
f(X1, X2, active(X3)) → f(X1, X2, X3)
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(z0, g(z0), z1)) → mark(f(z1, z1, z1))
active(g(b)) → mark(c)
active(b) → mark(c)
mark(f(z0, z1, z2)) → active(f(z0, z1, z2))
mark(g(z0)) → active(g(mark(z0)))
mark(b) → active(b)
mark(c) → active(c)
f(mark(z0), z1, z2) → f(z0, z1, z2)
f(z0, mark(z1), z2) → f(z0, z1, z2)
f(z0, z1, mark(z2)) → f(z0, z1, z2)
f(active(z0), z1, z2) → f(z0, z1, z2)
f(z0, active(z1), z2) → f(z0, z1, z2)
f(z0, z1, active(z2)) → f(z0, z1, z2)
g(mark(z0)) → g(z0)
g(active(z0)) → g(z0)
Tuples:
ACTIVE(f(z0, g(z0), z1)) → c1(MARK(f(z1, z1, z1)), F(z1, z1, z1))
ACTIVE(g(b)) → c2(MARK(c))
ACTIVE(b) → c3(MARK(c))
MARK(f(z0, z1, z2)) → c4(ACTIVE(f(z0, z1, z2)), F(z0, z1, z2))
MARK(g(z0)) → c5(ACTIVE(g(mark(z0))), G(mark(z0)), MARK(z0))
MARK(b) → c6(ACTIVE(b))
MARK(c) → c7(ACTIVE(c))
F(mark(z0), z1, z2) → c8(F(z0, z1, z2))
F(z0, mark(z1), z2) → c9(F(z0, z1, z2))
F(z0, z1, mark(z2)) → c10(F(z0, z1, z2))
F(active(z0), z1, z2) → c11(F(z0, z1, z2))
F(z0, active(z1), z2) → c12(F(z0, z1, z2))
F(z0, z1, active(z2)) → c13(F(z0, z1, z2))
G(mark(z0)) → c14(G(z0))
G(active(z0)) → c15(G(z0))
S tuples:
ACTIVE(f(z0, g(z0), z1)) → c1(MARK(f(z1, z1, z1)), F(z1, z1, z1))
ACTIVE(g(b)) → c2(MARK(c))
ACTIVE(b) → c3(MARK(c))
MARK(f(z0, z1, z2)) → c4(ACTIVE(f(z0, z1, z2)), F(z0, z1, z2))
MARK(g(z0)) → c5(ACTIVE(g(mark(z0))), G(mark(z0)), MARK(z0))
MARK(b) → c6(ACTIVE(b))
MARK(c) → c7(ACTIVE(c))
F(mark(z0), z1, z2) → c8(F(z0, z1, z2))
F(z0, mark(z1), z2) → c9(F(z0, z1, z2))
F(z0, z1, mark(z2)) → c10(F(z0, z1, z2))
F(active(z0), z1, z2) → c11(F(z0, z1, z2))
F(z0, active(z1), z2) → c12(F(z0, z1, z2))
F(z0, z1, active(z2)) → c13(F(z0, z1, z2))
G(mark(z0)) → c14(G(z0))
G(active(z0)) → c15(G(z0))
K tuples:none
Defined Rule Symbols:
active, mark, f, g
Defined Pair Symbols:
ACTIVE, MARK, F, G
Compound Symbols:
c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15
(3) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
ACTIVE(f(z0, g(z0), z1)) → c1(MARK(f(z1, z1, z1)), F(z1, z1, z1))
ACTIVE(g(b)) → c2(MARK(c))
MARK(f(z0, z1, z2)) → c4(ACTIVE(f(z0, z1, z2)), F(z0, z1, z2))
MARK(g(z0)) → c5(ACTIVE(g(mark(z0))), G(mark(z0)), MARK(z0))
F(mark(z0), z1, z2) → c8(F(z0, z1, z2))
F(z0, mark(z1), z2) → c9(F(z0, z1, z2))
F(z0, z1, mark(z2)) → c10(F(z0, z1, z2))
F(active(z0), z1, z2) → c11(F(z0, z1, z2))
F(z0, active(z1), z2) → c12(F(z0, z1, z2))
F(z0, z1, active(z2)) → c13(F(z0, z1, z2))
G(mark(z0)) → c14(G(z0))
G(active(z0)) → c15(G(z0))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(f(z0, g(z0), z1)) → mark(f(z1, z1, z1))
active(g(b)) → mark(c)
active(b) → mark(c)
mark(f(z0, z1, z2)) → active(f(z0, z1, z2))
mark(g(z0)) → active(g(mark(z0)))
mark(b) → active(b)
mark(c) → active(c)
f(mark(z0), z1, z2) → f(z0, z1, z2)
f(z0, mark(z1), z2) → f(z0, z1, z2)
f(z0, z1, mark(z2)) → f(z0, z1, z2)
f(active(z0), z1, z2) → f(z0, z1, z2)
f(z0, active(z1), z2) → f(z0, z1, z2)
f(z0, z1, active(z2)) → f(z0, z1, z2)
g(mark(z0)) → g(z0)
g(active(z0)) → g(z0)
Tuples:
ACTIVE(b) → c3(MARK(c))
MARK(b) → c6(ACTIVE(b))
MARK(c) → c7(ACTIVE(c))
S tuples:
ACTIVE(b) → c3(MARK(c))
MARK(b) → c6(ACTIVE(b))
MARK(c) → c7(ACTIVE(c))
K tuples:none
Defined Rule Symbols:
active, mark, f, g
Defined Pair Symbols:
ACTIVE, MARK
Compound Symbols:
c3, c6, c7
(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 3 trailing nodes:
ACTIVE(b) → c3(MARK(c))
MARK(c) → c7(ACTIVE(c))
MARK(b) → c6(ACTIVE(b))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(f(z0, g(z0), z1)) → mark(f(z1, z1, z1))
active(g(b)) → mark(c)
active(b) → mark(c)
mark(f(z0, z1, z2)) → active(f(z0, z1, z2))
mark(g(z0)) → active(g(mark(z0)))
mark(b) → active(b)
mark(c) → active(c)
f(mark(z0), z1, z2) → f(z0, z1, z2)
f(z0, mark(z1), z2) → f(z0, z1, z2)
f(z0, z1, mark(z2)) → f(z0, z1, z2)
f(active(z0), z1, z2) → f(z0, z1, z2)
f(z0, active(z1), z2) → f(z0, z1, z2)
f(z0, z1, active(z2)) → f(z0, z1, z2)
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))