(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(f(a, X, X)) → mark(f(X, b, b))
active(b) → mark(a)
mark(f(X1, X2, X3)) → active(f(X1, mark(X2), X3))
mark(a) → active(a)
mark(b) → active(b)
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)
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(f(a, z0, z0)) → mark(f(z0, b, b))
active(b) → mark(a)
mark(f(z0, z1, z2)) → active(f(z0, mark(z1), z2))
mark(a) → active(a)
mark(b) → active(b)
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)
Tuples:
ACTIVE(f(a, z0, z0)) → c(MARK(f(z0, b, b)), F(z0, b, b))
ACTIVE(b) → c1(MARK(a))
MARK(f(z0, z1, z2)) → c2(ACTIVE(f(z0, mark(z1), z2)), F(z0, mark(z1), z2), MARK(z1))
MARK(a) → c3(ACTIVE(a))
MARK(b) → c4(ACTIVE(b))
F(mark(z0), z1, z2) → c5(F(z0, z1, z2))
F(z0, mark(z1), z2) → c6(F(z0, z1, z2))
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2))
F(active(z0), z1, z2) → c8(F(z0, z1, z2))
F(z0, active(z1), z2) → c9(F(z0, z1, z2))
F(z0, z1, active(z2)) → c10(F(z0, z1, z2))
S tuples:
ACTIVE(f(a, z0, z0)) → c(MARK(f(z0, b, b)), F(z0, b, b))
ACTIVE(b) → c1(MARK(a))
MARK(f(z0, z1, z2)) → c2(ACTIVE(f(z0, mark(z1), z2)), F(z0, mark(z1), z2), MARK(z1))
MARK(a) → c3(ACTIVE(a))
MARK(b) → c4(ACTIVE(b))
F(mark(z0), z1, z2) → c5(F(z0, z1, z2))
F(z0, mark(z1), z2) → c6(F(z0, z1, z2))
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2))
F(active(z0), z1, z2) → c8(F(z0, z1, z2))
F(z0, active(z1), z2) → c9(F(z0, z1, z2))
F(z0, z1, active(z2)) → c10(F(z0, z1, z2))
K tuples:none
Defined Rule Symbols:
active, mark, f
Defined Pair Symbols:
ACTIVE, MARK, F
Compound Symbols:
c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10
(3) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
ACTIVE(f(a, z0, z0)) → c(MARK(f(z0, b, b)), F(z0, b, b))
MARK(f(z0, z1, z2)) → c2(ACTIVE(f(z0, mark(z1), z2)), F(z0, mark(z1), z2), MARK(z1))
F(mark(z0), z1, z2) → c5(F(z0, z1, z2))
F(z0, mark(z1), z2) → c6(F(z0, z1, z2))
F(z0, z1, mark(z2)) → c7(F(z0, z1, z2))
F(active(z0), z1, z2) → c8(F(z0, z1, z2))
F(z0, active(z1), z2) → c9(F(z0, z1, z2))
F(z0, z1, active(z2)) → c10(F(z0, z1, z2))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(f(a, z0, z0)) → mark(f(z0, b, b))
active(b) → mark(a)
mark(f(z0, z1, z2)) → active(f(z0, mark(z1), z2))
mark(a) → active(a)
mark(b) → active(b)
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)
Tuples:
ACTIVE(b) → c1(MARK(a))
MARK(a) → c3(ACTIVE(a))
MARK(b) → c4(ACTIVE(b))
S tuples:
ACTIVE(b) → c1(MARK(a))
MARK(a) → c3(ACTIVE(a))
MARK(b) → c4(ACTIVE(b))
K tuples:none
Defined Rule Symbols:
active, mark, f
Defined Pair Symbols:
ACTIVE, MARK
Compound Symbols:
c1, c3, c4
(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 3 trailing nodes:
MARK(b) → c4(ACTIVE(b))
MARK(a) → c3(ACTIVE(a))
ACTIVE(b) → c1(MARK(a))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(f(a, z0, z0)) → mark(f(z0, b, b))
active(b) → mark(a)
mark(f(z0, z1, z2)) → active(f(z0, mark(z1), z2))
mark(a) → active(a)
mark(b) → active(b)
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)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
active, mark, f
Defined Pair Symbols:none
Compound Symbols:none
(7) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(8) BOUNDS(O(1), O(1))