(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(f(X)) → mark(if(X, c, f(true)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
mark(f(X)) → active(f(mark(X)))
mark(if(X1, X2, X3)) → active(if(mark(X1), mark(X2), X3))
mark(c) → active(c)
mark(true) → active(true)
mark(false) → active(false)
f(mark(X)) → f(X)
f(active(X)) → f(X)
if(mark(X1), X2, X3) → if(X1, X2, X3)
if(X1, mark(X2), X3) → if(X1, X2, X3)
if(X1, X2, mark(X3)) → if(X1, X2, X3)
if(active(X1), X2, X3) → if(X1, X2, X3)
if(X1, active(X2), X3) → if(X1, X2, X3)
if(X1, X2, active(X3)) → if(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(z0)) → mark(if(z0, c, f(true)))
active(if(true, z0, z1)) → mark(z0)
active(if(false, z0, z1)) → mark(z1)
mark(f(z0)) → active(f(mark(z0)))
mark(if(z0, z1, z2)) → active(if(mark(z0), mark(z1), z2))
mark(c) → active(c)
mark(true) → active(true)
mark(false) → active(false)
f(mark(z0)) → f(z0)
f(active(z0)) → f(z0)
if(mark(z0), z1, z2) → if(z0, z1, z2)
if(z0, mark(z1), z2) → if(z0, z1, z2)
if(z0, z1, mark(z2)) → if(z0, z1, z2)
if(active(z0), z1, z2) → if(z0, z1, z2)
if(z0, active(z1), z2) → if(z0, z1, z2)
if(z0, z1, active(z2)) → if(z0, z1, z2)
Tuples:
ACTIVE(f(z0)) → c1(MARK(if(z0, c, f(true))), IF(z0, c, f(true)), F(true))
ACTIVE(if(true, z0, z1)) → c2(MARK(z0))
ACTIVE(if(false, z0, z1)) → c3(MARK(z1))
MARK(f(z0)) → c4(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(if(z0, z1, z2)) → c5(ACTIVE(if(mark(z0), mark(z1), z2)), IF(mark(z0), mark(z1), z2), MARK(z0), MARK(z1))
MARK(c) → c6(ACTIVE(c))
MARK(true) → c7(ACTIVE(true))
MARK(false) → c8(ACTIVE(false))
F(mark(z0)) → c9(F(z0))
F(active(z0)) → c10(F(z0))
IF(mark(z0), z1, z2) → c11(IF(z0, z1, z2))
IF(z0, mark(z1), z2) → c12(IF(z0, z1, z2))
IF(z0, z1, mark(z2)) → c13(IF(z0, z1, z2))
IF(active(z0), z1, z2) → c14(IF(z0, z1, z2))
IF(z0, active(z1), z2) → c15(IF(z0, z1, z2))
IF(z0, z1, active(z2)) → c16(IF(z0, z1, z2))
S tuples:
ACTIVE(f(z0)) → c1(MARK(if(z0, c, f(true))), IF(z0, c, f(true)), F(true))
ACTIVE(if(true, z0, z1)) → c2(MARK(z0))
ACTIVE(if(false, z0, z1)) → c3(MARK(z1))
MARK(f(z0)) → c4(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(if(z0, z1, z2)) → c5(ACTIVE(if(mark(z0), mark(z1), z2)), IF(mark(z0), mark(z1), z2), MARK(z0), MARK(z1))
MARK(c) → c6(ACTIVE(c))
MARK(true) → c7(ACTIVE(true))
MARK(false) → c8(ACTIVE(false))
F(mark(z0)) → c9(F(z0))
F(active(z0)) → c10(F(z0))
IF(mark(z0), z1, z2) → c11(IF(z0, z1, z2))
IF(z0, mark(z1), z2) → c12(IF(z0, z1, z2))
IF(z0, z1, mark(z2)) → c13(IF(z0, z1, z2))
IF(active(z0), z1, z2) → c14(IF(z0, z1, z2))
IF(z0, active(z1), z2) → c15(IF(z0, z1, z2))
IF(z0, z1, active(z2)) → c16(IF(z0, z1, z2))
K tuples:none
Defined Rule Symbols:
active, mark, f, if
Defined Pair Symbols:
ACTIVE, MARK, F, IF
Compound Symbols:
c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16
(3) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
ACTIVE(f(z0)) → c1(MARK(if(z0, c, f(true))), IF(z0, c, f(true)), F(true))
ACTIVE(if(true, z0, z1)) → c2(MARK(z0))
ACTIVE(if(false, z0, z1)) → c3(MARK(z1))
MARK(f(z0)) → c4(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0))
MARK(if(z0, z1, z2)) → c5(ACTIVE(if(mark(z0), mark(z1), z2)), IF(mark(z0), mark(z1), z2), MARK(z0), MARK(z1))
F(mark(z0)) → c9(F(z0))
F(active(z0)) → c10(F(z0))
IF(mark(z0), z1, z2) → c11(IF(z0, z1, z2))
IF(z0, mark(z1), z2) → c12(IF(z0, z1, z2))
IF(z0, z1, mark(z2)) → c13(IF(z0, z1, z2))
IF(active(z0), z1, z2) → c14(IF(z0, z1, z2))
IF(z0, active(z1), z2) → c15(IF(z0, z1, z2))
IF(z0, z1, active(z2)) → c16(IF(z0, z1, z2))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(f(z0)) → mark(if(z0, c, f(true)))
active(if(true, z0, z1)) → mark(z0)
active(if(false, z0, z1)) → mark(z1)
mark(f(z0)) → active(f(mark(z0)))
mark(if(z0, z1, z2)) → active(if(mark(z0), mark(z1), z2))
mark(c) → active(c)
mark(true) → active(true)
mark(false) → active(false)
f(mark(z0)) → f(z0)
f(active(z0)) → f(z0)
if(mark(z0), z1, z2) → if(z0, z1, z2)
if(z0, mark(z1), z2) → if(z0, z1, z2)
if(z0, z1, mark(z2)) → if(z0, z1, z2)
if(active(z0), z1, z2) → if(z0, z1, z2)
if(z0, active(z1), z2) → if(z0, z1, z2)
if(z0, z1, active(z2)) → if(z0, z1, z2)
Tuples:
MARK(c) → c6(ACTIVE(c))
MARK(true) → c7(ACTIVE(true))
MARK(false) → c8(ACTIVE(false))
S tuples:
MARK(c) → c6(ACTIVE(c))
MARK(true) → c7(ACTIVE(true))
MARK(false) → c8(ACTIVE(false))
K tuples:none
Defined Rule Symbols:
active, mark, f, if
Defined Pair Symbols:
MARK
Compound Symbols:
c6, c7, c8
(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 3 trailing nodes:
MARK(false) → c8(ACTIVE(false))
MARK(true) → c7(ACTIVE(true))
MARK(c) → c6(ACTIVE(c))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(f(z0)) → mark(if(z0, c, f(true)))
active(if(true, z0, z1)) → mark(z0)
active(if(false, z0, z1)) → mark(z1)
mark(f(z0)) → active(f(mark(z0)))
mark(if(z0, z1, z2)) → active(if(mark(z0), mark(z1), z2))
mark(c) → active(c)
mark(true) → active(true)
mark(false) → active(false)
f(mark(z0)) → f(z0)
f(active(z0)) → f(z0)
if(mark(z0), z1, z2) → if(z0, z1, z2)
if(z0, mark(z1), z2) → if(z0, z1, z2)
if(z0, z1, mark(z2)) → if(z0, z1, z2)
if(active(z0), z1, z2) → if(z0, z1, z2)
if(z0, active(z1), z2) → if(z0, z1, z2)
if(z0, z1, active(z2)) → if(z0, z1, z2)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
active, mark, f, if
Defined Pair Symbols:none
Compound Symbols:none
(7) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(8) BOUNDS(O(1), O(1))