Runtime Complexity (innermost) proof of /tmp/tmp7azXVz/Ex5_Zan97

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^1)).

0 CpxTRS

↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)

↳2 CdtProblem

    ↳3 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳4 CdtProblem

    ↳5 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 920 ms)

    ↳6 CdtProblem

    ↳7 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳8 CdtProblem

    ↳9 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳10 CdtProblem

    ↳11 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳12 CdtProblem

    ↳13 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 386 ms)

    ↳14 CdtProblem

    ↳15 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳16 CdtProblem

    ↳17 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 269 ms)

    ↳18 CdtProblem

    ↳19 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 940 ms)

    ↳20 CdtProblem

    ↳21 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 3 ms)

    ↳22 CdtProblem

    ↳23 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 403 ms)

    ↳24 CdtProblem

    ↳25 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 11 ms)

    ↳26 CdtProblem

    ↳27 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 1905 ms)

    ↳28 CdtProblem

    ↳29 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳30 CdtProblem

↳31 CpxTrsMatchBoundsTAProof (⇔, 0 ms)

↳32 BOUNDS(O(1), O(n^1))

(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) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 3 trailing nodes:

MARK(c) → c6(ACTIVE(c))
MARK(true) → c7(ACTIVE(true))
MARK(false) → c8(ACTIVE(false))

(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:

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))

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))
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, c9, c10, c11, c12, c13, c14, c15, c16

(5) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

ACTIVE(if(false, z0, z1)) → c3(MARK(z1))

We considered the (Usable) Rules:

mark(if(z0, z1, z2)) → active(if(mark(z0), mark(z1), z2))
mark(f(z0)) → active(f(mark(z0)))
mark(c) → active(c)
mark(true) → active(true)
mark(false) → active(false)
active(f(z0)) → mark(if(z0, c, f(true)))
active(if(true, z0, z1)) → mark(z0)
active(if(false, z0, z1)) → mark(z1)
if(z0, mark(z1), z2) → if(z0, z1, z2)
if(mark(z0), 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)
f(active(z0)) → f(z0)
f(mark(z0)) → f(z0)

And the 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))
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))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVE(x₁)) = [2]x₁
POL(F(x₁)) = 0
POL(IF(x₁, x₂, x₃)) = 0
POL(MARK(x₁)) = [4]x₁
POL(active(x₁)) = x₁
POL(c) = 0
POL(c1(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c4(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c5(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(c9(x₁)) = x₁
POL(f(x₁)) = [5]x₁
POL(false) = [2]
POL(if(x₁, x₂, x₃)) = [2]x₁ + [4]x₂ + [2]x₃
POL(mark(x₁)) = x₁
POL(true) = 0

(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:

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))

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))
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))

K tuples:

ACTIVE(if(false, z0, z1)) → c3(MARK(z1))

Defined Rule Symbols:

active, mark, f, if

Defined Pair Symbols:

ACTIVE, MARK, F, IF

Compound Symbols:

c1, c2, c3, c4, c5, c9, c10, c11, c12, c13, c14, c15, c16

(7) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace ACTIVE(f(z0)) → c1(MARK(if(z0, c, f(true))), IF(z0, c, f(true)), F(true)) by

ACTIVE(f(x0)) → c1(MARK(if(x0, c, f(true))), IF(x0, c, f(true)))

(8) 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(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))
ACTIVE(f(x0)) → c1(MARK(if(x0, c, f(true))), IF(x0, c, f(true)))

S tuples:

ACTIVE(if(true, z0, z1)) → c2(MARK(z0))
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))
ACTIVE(f(x0)) → c1(MARK(if(x0, c, f(true))), IF(x0, c, f(true)))

K tuples:

ACTIVE(if(false, z0, z1)) → c3(MARK(z1))

Defined Rule Symbols:

active, mark, f, if

Defined Pair Symbols:

ACTIVE, MARK, F, IF

Compound Symbols:

c2, c3, c4, c5, c9, c10, c11, c12, c13, c14, c15, c16, c1

(9) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace MARK(f(z0)) → c4(ACTIVE(f(mark(z0))), F(mark(z0)), MARK(z0)) by

MARK(f(z0)) → c4(ACTIVE(f(z0)), F(mark(z0)), MARK(z0))
MARK(f(f(z0))) → c4(ACTIVE(f(active(f(mark(z0))))), F(mark(f(z0))), MARK(f(z0)))
MARK(f(if(z0, z1, z2))) → c4(ACTIVE(f(active(if(mark(z0), mark(z1), z2)))), F(mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
MARK(f(c)) → c4(ACTIVE(f(active(c))), F(mark(c)), MARK(c))
MARK(f(true)) → c4(ACTIVE(f(active(true))), F(mark(true)), MARK(true))
MARK(f(false)) → c4(ACTIVE(f(active(false))), F(mark(false)), MARK(false))
MARK(f(x0)) → c4

(10) 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(if(true, z0, z1)) → c2(MARK(z0))
ACTIVE(if(false, z0, z1)) → c3(MARK(z1))
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))
ACTIVE(f(x0)) → c1(MARK(if(x0, c, f(true))), IF(x0, c, f(true)))
MARK(f(z0)) → c4(ACTIVE(f(z0)), F(mark(z0)), MARK(z0))
MARK(f(f(z0))) → c4(ACTIVE(f(active(f(mark(z0))))), F(mark(f(z0))), MARK(f(z0)))
MARK(f(if(z0, z1, z2))) → c4(ACTIVE(f(active(if(mark(z0), mark(z1), z2)))), F(mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
MARK(f(c)) → c4(ACTIVE(f(active(c))), F(mark(c)), MARK(c))
MARK(f(true)) → c4(ACTIVE(f(active(true))), F(mark(true)), MARK(true))
MARK(f(false)) → c4(ACTIVE(f(active(false))), F(mark(false)), MARK(false))
MARK(f(x0)) → c4

S tuples:

ACTIVE(if(true, z0, z1)) → c2(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))
ACTIVE(f(x0)) → c1(MARK(if(x0, c, f(true))), IF(x0, c, f(true)))
MARK(f(z0)) → c4(ACTIVE(f(z0)), F(mark(z0)), MARK(z0))
MARK(f(f(z0))) → c4(ACTIVE(f(active(f(mark(z0))))), F(mark(f(z0))), MARK(f(z0)))
MARK(f(if(z0, z1, z2))) → c4(ACTIVE(f(active(if(mark(z0), mark(z1), z2)))), F(mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
MARK(f(c)) → c4(ACTIVE(f(active(c))), F(mark(c)), MARK(c))
MARK(f(true)) → c4(ACTIVE(f(active(true))), F(mark(true)), MARK(true))
MARK(f(false)) → c4(ACTIVE(f(active(false))), F(mark(false)), MARK(false))
MARK(f(x0)) → c4

K tuples:

ACTIVE(if(false, z0, z1)) → c3(MARK(z1))

Defined Rule Symbols:

active, mark, f, if

Defined Pair Symbols:

ACTIVE, MARK, F, IF

Compound Symbols:

c2, c3, c5, c9, c10, c11, c12, c13, c14, c15, c16, c1, c4, c4

(11) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing nodes:

MARK(f(x0)) → c4

(12) 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(if(true, z0, z1)) → c2(MARK(z0))
ACTIVE(if(false, z0, z1)) → c3(MARK(z1))
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))
ACTIVE(f(x0)) → c1(MARK(if(x0, c, f(true))), IF(x0, c, f(true)))
MARK(f(z0)) → c4(ACTIVE(f(z0)), F(mark(z0)), MARK(z0))
MARK(f(f(z0))) → c4(ACTIVE(f(active(f(mark(z0))))), F(mark(f(z0))), MARK(f(z0)))
MARK(f(if(z0, z1, z2))) → c4(ACTIVE(f(active(if(mark(z0), mark(z1), z2)))), F(mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
MARK(f(c)) → c4(ACTIVE(f(active(c))), F(mark(c)), MARK(c))
MARK(f(true)) → c4(ACTIVE(f(active(true))), F(mark(true)), MARK(true))
MARK(f(false)) → c4(ACTIVE(f(active(false))), F(mark(false)), MARK(false))

S tuples:

ACTIVE(if(true, z0, z1)) → c2(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))
ACTIVE(f(x0)) → c1(MARK(if(x0, c, f(true))), IF(x0, c, f(true)))
MARK(f(z0)) → c4(ACTIVE(f(z0)), F(mark(z0)), MARK(z0))
MARK(f(f(z0))) → c4(ACTIVE(f(active(f(mark(z0))))), F(mark(f(z0))), MARK(f(z0)))
MARK(f(if(z0, z1, z2))) → c4(ACTIVE(f(active(if(mark(z0), mark(z1), z2)))), F(mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
MARK(f(c)) → c4(ACTIVE(f(active(c))), F(mark(c)), MARK(c))
MARK(f(true)) → c4(ACTIVE(f(active(true))), F(mark(true)), MARK(true))
MARK(f(false)) → c4(ACTIVE(f(active(false))), F(mark(false)), MARK(false))

K tuples:

ACTIVE(if(false, z0, z1)) → c3(MARK(z1))

Defined Rule Symbols:

active, mark, f, if

Defined Pair Symbols:

ACTIVE, MARK, F, IF

Compound Symbols:

c2, c3, c5, c9, c10, c11, c12, c13, c14, c15, c16, c1, c4

(13) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

MARK(f(false)) → c4(ACTIVE(f(active(false))), F(mark(false)), MARK(false))

We considered the (Usable) Rules:

f(active(z0)) → f(z0)
f(mark(z0)) → f(z0)
mark(false) → active(false)
mark(true) → active(true)
mark(c) → active(c)
mark(if(z0, z1, z2)) → active(if(mark(z0), mark(z1), z2))
mark(f(z0)) → active(f(mark(z0)))
active(f(z0)) → mark(if(z0, c, f(true)))
active(if(true, z0, z1)) → mark(z0)
active(if(false, z0, z1)) → mark(z1)
if(z0, mark(z1), z2) → if(z0, z1, z2)
if(mark(z0), 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)

And the Tuples:

ACTIVE(if(true, z0, z1)) → c2(MARK(z0))
ACTIVE(if(false, z0, z1)) → c3(MARK(z1))
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))
ACTIVE(f(x0)) → c1(MARK(if(x0, c, f(true))), IF(x0, c, f(true)))
MARK(f(z0)) → c4(ACTIVE(f(z0)), F(mark(z0)), MARK(z0))
MARK(f(f(z0))) → c4(ACTIVE(f(active(f(mark(z0))))), F(mark(f(z0))), MARK(f(z0)))
MARK(f(if(z0, z1, z2))) → c4(ACTIVE(f(active(if(mark(z0), mark(z1), z2)))), F(mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
MARK(f(c)) → c4(ACTIVE(f(active(c))), F(mark(c)), MARK(c))
MARK(f(true)) → c4(ACTIVE(f(active(true))), F(mark(true)), MARK(true))
MARK(f(false)) → c4(ACTIVE(f(active(false))), F(mark(false)), MARK(false))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVE(x₁)) = x₁
POL(F(x₁)) = 0
POL(IF(x₁, x₂, x₃)) = 0
POL(MARK(x₁)) = [2]x₁
POL(active(x₁)) = x₁
POL(c) = 0
POL(c1(x₁, x₂)) = x₁ + x₂
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c4(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c5(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(c9(x₁)) = x₁
POL(f(x₁)) = [4]x₁
POL(false) = [2]
POL(if(x₁, x₂, x₃)) = [2]x₁ + [2]x₂ + [2]x₃
POL(mark(x₁)) = x₁
POL(true) = 0

(14) 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(if(true, z0, z1)) → c2(MARK(z0))
ACTIVE(if(false, z0, z1)) → c3(MARK(z1))
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))
ACTIVE(f(x0)) → c1(MARK(if(x0, c, f(true))), IF(x0, c, f(true)))
MARK(f(z0)) → c4(ACTIVE(f(z0)), F(mark(z0)), MARK(z0))
MARK(f(f(z0))) → c4(ACTIVE(f(active(f(mark(z0))))), F(mark(f(z0))), MARK(f(z0)))
MARK(f(if(z0, z1, z2))) → c4(ACTIVE(f(active(if(mark(z0), mark(z1), z2)))), F(mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
MARK(f(c)) → c4(ACTIVE(f(active(c))), F(mark(c)), MARK(c))
MARK(f(true)) → c4(ACTIVE(f(active(true))), F(mark(true)), MARK(true))
MARK(f(false)) → c4(ACTIVE(f(active(false))), F(mark(false)), MARK(false))

S tuples:

ACTIVE(if(true, z0, z1)) → c2(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))
ACTIVE(f(x0)) → c1(MARK(if(x0, c, f(true))), IF(x0, c, f(true)))
MARK(f(z0)) → c4(ACTIVE(f(z0)), F(mark(z0)), MARK(z0))
MARK(f(f(z0))) → c4(ACTIVE(f(active(f(mark(z0))))), F(mark(f(z0))), MARK(f(z0)))
MARK(f(if(z0, z1, z2))) → c4(ACTIVE(f(active(if(mark(z0), mark(z1), z2)))), F(mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
MARK(f(c)) → c4(ACTIVE(f(active(c))), F(mark(c)), MARK(c))
MARK(f(true)) → c4(ACTIVE(f(active(true))), F(mark(true)), MARK(true))

K tuples:

ACTIVE(if(false, z0, z1)) → c3(MARK(z1))
MARK(f(false)) → c4(ACTIVE(f(active(false))), F(mark(false)), MARK(false))

Defined Rule Symbols:

active, mark, f, if

Defined Pair Symbols:

ACTIVE, MARK, F, IF

Compound Symbols:

c2, c3, c5, c9, c10, c11, c12, c13, c14, c15, c16, c1, c4

(15) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace MARK(if(z0, z1, z2)) → c5(ACTIVE(if(mark(z0), mark(z1), z2)), IF(mark(z0), mark(z1), z2), MARK(z0), MARK(z1)) by

MARK(if(z0, x1, z2)) → c5(ACTIVE(if(z0, mark(x1), z2)), IF(mark(z0), mark(x1), z2), MARK(z0), MARK(x1))
MARK(if(x0, z1, z2)) → c5(ACTIVE(if(mark(x0), z1, z2)), IF(mark(x0), mark(z1), z2), MARK(x0), MARK(z1))
MARK(if(x0, f(z0), x2)) → c5(ACTIVE(if(mark(x0), active(f(mark(z0))), x2)), IF(mark(x0), mark(f(z0)), x2), MARK(x0), MARK(f(z0)))
MARK(if(x0, if(z0, z1, z2), x2)) → c5(ACTIVE(if(mark(x0), active(if(mark(z0), mark(z1), z2)), x2)), IF(mark(x0), mark(if(z0, z1, z2)), x2), MARK(x0), MARK(if(z0, z1, z2)))
MARK(if(x0, c, x2)) → c5(ACTIVE(if(mark(x0), active(c), x2)), IF(mark(x0), mark(c), x2), MARK(x0), MARK(c))
MARK(if(x0, true, x2)) → c5(ACTIVE(if(mark(x0), active(true), x2)), IF(mark(x0), mark(true), x2), MARK(x0), MARK(true))
MARK(if(x0, false, x2)) → c5(ACTIVE(if(mark(x0), active(false), x2)), IF(mark(x0), mark(false), x2), MARK(x0), MARK(false))
MARK(if(f(z0), x1, x2)) → c5(ACTIVE(if(active(f(mark(z0))), mark(x1), x2)), IF(mark(f(z0)), mark(x1), x2), MARK(f(z0)), MARK(x1))
MARK(if(if(z0, z1, z2), x1, x2)) → c5(ACTIVE(if(active(if(mark(z0), mark(z1), z2)), mark(x1), x2)), IF(mark(if(z0, z1, z2)), mark(x1), x2), MARK(if(z0, z1, z2)), MARK(x1))
MARK(if(c, x1, x2)) → c5(ACTIVE(if(active(c), mark(x1), x2)), IF(mark(c), mark(x1), x2), MARK(c), MARK(x1))
MARK(if(true, x1, x2)) → c5(ACTIVE(if(active(true), mark(x1), x2)), IF(mark(true), mark(x1), x2), MARK(true), MARK(x1))
MARK(if(false, x1, x2)) → c5(ACTIVE(if(active(false), mark(x1), x2)), IF(mark(false), mark(x1), x2), MARK(false), MARK(x1))
MARK(if(x0, x1, x2)) → c5(IF(mark(x0), mark(x1), x2))

(16) 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(if(true, z0, z1)) → c2(MARK(z0))
ACTIVE(if(false, z0, z1)) → c3(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))
ACTIVE(f(x0)) → c1(MARK(if(x0, c, f(true))), IF(x0, c, f(true)))
MARK(f(z0)) → c4(ACTIVE(f(z0)), F(mark(z0)), MARK(z0))
MARK(f(f(z0))) → c4(ACTIVE(f(active(f(mark(z0))))), F(mark(f(z0))), MARK(f(z0)))
MARK(f(if(z0, z1, z2))) → c4(ACTIVE(f(active(if(mark(z0), mark(z1), z2)))), F(mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
MARK(f(c)) → c4(ACTIVE(f(active(c))), F(mark(c)), MARK(c))
MARK(f(true)) → c4(ACTIVE(f(active(true))), F(mark(true)), MARK(true))
MARK(f(false)) → c4(ACTIVE(f(active(false))), F(mark(false)), MARK(false))
MARK(if(z0, x1, z2)) → c5(ACTIVE(if(z0, mark(x1), z2)), IF(mark(z0), mark(x1), z2), MARK(z0), MARK(x1))
MARK(if(x0, z1, z2)) → c5(ACTIVE(if(mark(x0), z1, z2)), IF(mark(x0), mark(z1), z2), MARK(x0), MARK(z1))
MARK(if(x0, f(z0), x2)) → c5(ACTIVE(if(mark(x0), active(f(mark(z0))), x2)), IF(mark(x0), mark(f(z0)), x2), MARK(x0), MARK(f(z0)))
MARK(if(x0, if(z0, z1, z2), x2)) → c5(ACTIVE(if(mark(x0), active(if(mark(z0), mark(z1), z2)), x2)), IF(mark(x0), mark(if(z0, z1, z2)), x2), MARK(x0), MARK(if(z0, z1, z2)))
MARK(if(x0, c, x2)) → c5(ACTIVE(if(mark(x0), active(c), x2)), IF(mark(x0), mark(c), x2), MARK(x0), MARK(c))
MARK(if(x0, true, x2)) → c5(ACTIVE(if(mark(x0), active(true), x2)), IF(mark(x0), mark(true), x2), MARK(x0), MARK(true))
MARK(if(x0, false, x2)) → c5(ACTIVE(if(mark(x0), active(false), x2)), IF(mark(x0), mark(false), x2), MARK(x0), MARK(false))
MARK(if(f(z0), x1, x2)) → c5(ACTIVE(if(active(f(mark(z0))), mark(x1), x2)), IF(mark(f(z0)), mark(x1), x2), MARK(f(z0)), MARK(x1))
MARK(if(if(z0, z1, z2), x1, x2)) → c5(ACTIVE(if(active(if(mark(z0), mark(z1), z2)), mark(x1), x2)), IF(mark(if(z0, z1, z2)), mark(x1), x2), MARK(if(z0, z1, z2)), MARK(x1))
MARK(if(c, x1, x2)) → c5(ACTIVE(if(active(c), mark(x1), x2)), IF(mark(c), mark(x1), x2), MARK(c), MARK(x1))
MARK(if(true, x1, x2)) → c5(ACTIVE(if(active(true), mark(x1), x2)), IF(mark(true), mark(x1), x2), MARK(true), MARK(x1))
MARK(if(false, x1, x2)) → c5(ACTIVE(if(active(false), mark(x1), x2)), IF(mark(false), mark(x1), x2), MARK(false), MARK(x1))
MARK(if(x0, x1, x2)) → c5(IF(mark(x0), mark(x1), x2))

S tuples:

ACTIVE(if(true, z0, z1)) → c2(MARK(z0))
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))
ACTIVE(f(x0)) → c1(MARK(if(x0, c, f(true))), IF(x0, c, f(true)))
MARK(f(z0)) → c4(ACTIVE(f(z0)), F(mark(z0)), MARK(z0))
MARK(f(f(z0))) → c4(ACTIVE(f(active(f(mark(z0))))), F(mark(f(z0))), MARK(f(z0)))
MARK(f(if(z0, z1, z2))) → c4(ACTIVE(f(active(if(mark(z0), mark(z1), z2)))), F(mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
MARK(f(c)) → c4(ACTIVE(f(active(c))), F(mark(c)), MARK(c))
MARK(f(true)) → c4(ACTIVE(f(active(true))), F(mark(true)), MARK(true))
MARK(if(z0, x1, z2)) → c5(ACTIVE(if(z0, mark(x1), z2)), IF(mark(z0), mark(x1), z2), MARK(z0), MARK(x1))
MARK(if(x0, z1, z2)) → c5(ACTIVE(if(mark(x0), z1, z2)), IF(mark(x0), mark(z1), z2), MARK(x0), MARK(z1))
MARK(if(x0, f(z0), x2)) → c5(ACTIVE(if(mark(x0), active(f(mark(z0))), x2)), IF(mark(x0), mark(f(z0)), x2), MARK(x0), MARK(f(z0)))
MARK(if(x0, if(z0, z1, z2), x2)) → c5(ACTIVE(if(mark(x0), active(if(mark(z0), mark(z1), z2)), x2)), IF(mark(x0), mark(if(z0, z1, z2)), x2), MARK(x0), MARK(if(z0, z1, z2)))
MARK(if(x0, c, x2)) → c5(ACTIVE(if(mark(x0), active(c), x2)), IF(mark(x0), mark(c), x2), MARK(x0), MARK(c))
MARK(if(x0, true, x2)) → c5(ACTIVE(if(mark(x0), active(true), x2)), IF(mark(x0), mark(true), x2), MARK(x0), MARK(true))
MARK(if(x0, false, x2)) → c5(ACTIVE(if(mark(x0), active(false), x2)), IF(mark(x0), mark(false), x2), MARK(x0), MARK(false))
MARK(if(f(z0), x1, x2)) → c5(ACTIVE(if(active(f(mark(z0))), mark(x1), x2)), IF(mark(f(z0)), mark(x1), x2), MARK(f(z0)), MARK(x1))
MARK(if(if(z0, z1, z2), x1, x2)) → c5(ACTIVE(if(active(if(mark(z0), mark(z1), z2)), mark(x1), x2)), IF(mark(if(z0, z1, z2)), mark(x1), x2), MARK(if(z0, z1, z2)), MARK(x1))
MARK(if(c, x1, x2)) → c5(ACTIVE(if(active(c), mark(x1), x2)), IF(mark(c), mark(x1), x2), MARK(c), MARK(x1))
MARK(if(true, x1, x2)) → c5(ACTIVE(if(active(true), mark(x1), x2)), IF(mark(true), mark(x1), x2), MARK(true), MARK(x1))
MARK(if(false, x1, x2)) → c5(ACTIVE(if(active(false), mark(x1), x2)), IF(mark(false), mark(x1), x2), MARK(false), MARK(x1))
MARK(if(x0, x1, x2)) → c5(IF(mark(x0), mark(x1), x2))

K tuples:

ACTIVE(if(false, z0, z1)) → c3(MARK(z1))
MARK(f(false)) → c4(ACTIVE(f(active(false))), F(mark(false)), MARK(false))

Defined Rule Symbols:

active, mark, f, if

Defined Pair Symbols:

ACTIVE, F, IF, MARK

Compound Symbols:

c2, c3, c9, c10, c11, c12, c13, c14, c15, c16, c1, c4, c5, c5

(17) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

MARK(if(x0, false, x2)) → c5(ACTIVE(if(mark(x0), active(false), x2)), IF(mark(x0), mark(false), x2), MARK(x0), MARK(false))

We considered the (Usable) Rules:

mark(if(z0, z1, z2)) → active(if(mark(z0), mark(z1), z2))
mark(f(z0)) → active(f(mark(z0)))
mark(c) → active(c)
mark(true) → active(true)
mark(false) → active(false)
active(f(z0)) → mark(if(z0, c, f(true)))
active(if(true, z0, z1)) → mark(z0)
active(if(false, z0, z1)) → mark(z1)
if(z0, mark(z1), z2) → if(z0, z1, z2)
if(mark(z0), 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)
f(active(z0)) → f(z0)
f(mark(z0)) → f(z0)

And the Tuples:

ACTIVE(if(true, z0, z1)) → c2(MARK(z0))
ACTIVE(if(false, z0, z1)) → c3(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))
ACTIVE(f(x0)) → c1(MARK(if(x0, c, f(true))), IF(x0, c, f(true)))
MARK(f(z0)) → c4(ACTIVE(f(z0)), F(mark(z0)), MARK(z0))
MARK(f(f(z0))) → c4(ACTIVE(f(active(f(mark(z0))))), F(mark(f(z0))), MARK(f(z0)))
MARK(f(if(z0, z1, z2))) → c4(ACTIVE(f(active(if(mark(z0), mark(z1), z2)))), F(mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
MARK(f(c)) → c4(ACTIVE(f(active(c))), F(mark(c)), MARK(c))
MARK(f(true)) → c4(ACTIVE(f(active(true))), F(mark(true)), MARK(true))
MARK(f(false)) → c4(ACTIVE(f(active(false))), F(mark(false)), MARK(false))
MARK(if(z0, x1, z2)) → c5(ACTIVE(if(z0, mark(x1), z2)), IF(mark(z0), mark(x1), z2), MARK(z0), MARK(x1))
MARK(if(x0, z1, z2)) → c5(ACTIVE(if(mark(x0), z1, z2)), IF(mark(x0), mark(z1), z2), MARK(x0), MARK(z1))
MARK(if(x0, f(z0), x2)) → c5(ACTIVE(if(mark(x0), active(f(mark(z0))), x2)), IF(mark(x0), mark(f(z0)), x2), MARK(x0), MARK(f(z0)))
MARK(if(x0, if(z0, z1, z2), x2)) → c5(ACTIVE(if(mark(x0), active(if(mark(z0), mark(z1), z2)), x2)), IF(mark(x0), mark(if(z0, z1, z2)), x2), MARK(x0), MARK(if(z0, z1, z2)))
MARK(if(x0, c, x2)) → c5(ACTIVE(if(mark(x0), active(c), x2)), IF(mark(x0), mark(c), x2), MARK(x0), MARK(c))
MARK(if(x0, true, x2)) → c5(ACTIVE(if(mark(x0), active(true), x2)), IF(mark(x0), mark(true), x2), MARK(x0), MARK(true))
MARK(if(x0, false, x2)) → c5(ACTIVE(if(mark(x0), active(false), x2)), IF(mark(x0), mark(false), x2), MARK(x0), MARK(false))
MARK(if(f(z0), x1, x2)) → c5(ACTIVE(if(active(f(mark(z0))), mark(x1), x2)), IF(mark(f(z0)), mark(x1), x2), MARK(f(z0)), MARK(x1))
MARK(if(if(z0, z1, z2), x1, x2)) → c5(ACTIVE(if(active(if(mark(z0), mark(z1), z2)), mark(x1), x2)), IF(mark(if(z0, z1, z2)), mark(x1), x2), MARK(if(z0, z1, z2)), MARK(x1))
MARK(if(c, x1, x2)) → c5(ACTIVE(if(active(c), mark(x1), x2)), IF(mark(c), mark(x1), x2), MARK(c), MARK(x1))
MARK(if(true, x1, x2)) → c5(ACTIVE(if(active(true), mark(x1), x2)), IF(mark(true), mark(x1), x2), MARK(true), MARK(x1))
MARK(if(false, x1, x2)) → c5(ACTIVE(if(active(false), mark(x1), x2)), IF(mark(false), mark(x1), x2), MARK(false), MARK(x1))
MARK(if(x0, x1, x2)) → c5(IF(mark(x0), mark(x1), x2))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVE(x₁)) = [2]x₁
POL(F(x₁)) = 0
POL(IF(x₁, x₂, x₃)) = [3]x₂ + [2]x₃
POL(MARK(x₁)) = [4]x₁
POL(active(x₁)) = x₁
POL(c) = 0
POL(c1(x₁, x₂)) = x₁ + x₂
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c4(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c5(x₁)) = x₁
POL(c5(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(c9(x₁)) = x₁
POL(f(x₁)) = [4]x₁
POL(false) = [1]
POL(if(x₁, x₂, x₃)) = [2]x₁ + [5]x₂ + [2]x₃
POL(mark(x₁)) = x₁
POL(true) = 0

(18) 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(if(true, z0, z1)) → c2(MARK(z0))
ACTIVE(if(false, z0, z1)) → c3(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))
ACTIVE(f(x0)) → c1(MARK(if(x0, c, f(true))), IF(x0, c, f(true)))
MARK(f(z0)) → c4(ACTIVE(f(z0)), F(mark(z0)), MARK(z0))
MARK(f(f(z0))) → c4(ACTIVE(f(active(f(mark(z0))))), F(mark(f(z0))), MARK(f(z0)))
MARK(f(if(z0, z1, z2))) → c4(ACTIVE(f(active(if(mark(z0), mark(z1), z2)))), F(mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
MARK(f(c)) → c4(ACTIVE(f(active(c))), F(mark(c)), MARK(c))
MARK(f(true)) → c4(ACTIVE(f(active(true))), F(mark(true)), MARK(true))
MARK(f(false)) → c4(ACTIVE(f(active(false))), F(mark(false)), MARK(false))
MARK(if(z0, x1, z2)) → c5(ACTIVE(if(z0, mark(x1), z2)), IF(mark(z0), mark(x1), z2), MARK(z0), MARK(x1))
MARK(if(x0, z1, z2)) → c5(ACTIVE(if(mark(x0), z1, z2)), IF(mark(x0), mark(z1), z2), MARK(x0), MARK(z1))
MARK(if(x0, f(z0), x2)) → c5(ACTIVE(if(mark(x0), active(f(mark(z0))), x2)), IF(mark(x0), mark(f(z0)), x2), MARK(x0), MARK(f(z0)))
MARK(if(x0, if(z0, z1, z2), x2)) → c5(ACTIVE(if(mark(x0), active(if(mark(z0), mark(z1), z2)), x2)), IF(mark(x0), mark(if(z0, z1, z2)), x2), MARK(x0), MARK(if(z0, z1, z2)))
MARK(if(x0, c, x2)) → c5(ACTIVE(if(mark(x0), active(c), x2)), IF(mark(x0), mark(c), x2), MARK(x0), MARK(c))
MARK(if(x0, true, x2)) → c5(ACTIVE(if(mark(x0), active(true), x2)), IF(mark(x0), mark(true), x2), MARK(x0), MARK(true))
MARK(if(x0, false, x2)) → c5(ACTIVE(if(mark(x0), active(false), x2)), IF(mark(x0), mark(false), x2), MARK(x0), MARK(false))
MARK(if(f(z0), x1, x2)) → c5(ACTIVE(if(active(f(mark(z0))), mark(x1), x2)), IF(mark(f(z0)), mark(x1), x2), MARK(f(z0)), MARK(x1))
MARK(if(if(z0, z1, z2), x1, x2)) → c5(ACTIVE(if(active(if(mark(z0), mark(z1), z2)), mark(x1), x2)), IF(mark(if(z0, z1, z2)), mark(x1), x2), MARK(if(z0, z1, z2)), MARK(x1))
MARK(if(c, x1, x2)) → c5(ACTIVE(if(active(c), mark(x1), x2)), IF(mark(c), mark(x1), x2), MARK(c), MARK(x1))
MARK(if(true, x1, x2)) → c5(ACTIVE(if(active(true), mark(x1), x2)), IF(mark(true), mark(x1), x2), MARK(true), MARK(x1))
MARK(if(false, x1, x2)) → c5(ACTIVE(if(active(false), mark(x1), x2)), IF(mark(false), mark(x1), x2), MARK(false), MARK(x1))
MARK(if(x0, x1, x2)) → c5(IF(mark(x0), mark(x1), x2))

S tuples:

ACTIVE(if(true, z0, z1)) → c2(MARK(z0))
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))
ACTIVE(f(x0)) → c1(MARK(if(x0, c, f(true))), IF(x0, c, f(true)))
MARK(f(z0)) → c4(ACTIVE(f(z0)), F(mark(z0)), MARK(z0))
MARK(f(f(z0))) → c4(ACTIVE(f(active(f(mark(z0))))), F(mark(f(z0))), MARK(f(z0)))
MARK(f(if(z0, z1, z2))) → c4(ACTIVE(f(active(if(mark(z0), mark(z1), z2)))), F(mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
MARK(f(c)) → c4(ACTIVE(f(active(c))), F(mark(c)), MARK(c))
MARK(f(true)) → c4(ACTIVE(f(active(true))), F(mark(true)), MARK(true))
MARK(if(z0, x1, z2)) → c5(ACTIVE(if(z0, mark(x1), z2)), IF(mark(z0), mark(x1), z2), MARK(z0), MARK(x1))
MARK(if(x0, z1, z2)) → c5(ACTIVE(if(mark(x0), z1, z2)), IF(mark(x0), mark(z1), z2), MARK(x0), MARK(z1))
MARK(if(x0, f(z0), x2)) → c5(ACTIVE(if(mark(x0), active(f(mark(z0))), x2)), IF(mark(x0), mark(f(z0)), x2), MARK(x0), MARK(f(z0)))
MARK(if(x0, if(z0, z1, z2), x2)) → c5(ACTIVE(if(mark(x0), active(if(mark(z0), mark(z1), z2)), x2)), IF(mark(x0), mark(if(z0, z1, z2)), x2), MARK(x0), MARK(if(z0, z1, z2)))
MARK(if(x0, c, x2)) → c5(ACTIVE(if(mark(x0), active(c), x2)), IF(mark(x0), mark(c), x2), MARK(x0), MARK(c))
MARK(if(x0, true, x2)) → c5(ACTIVE(if(mark(x0), active(true), x2)), IF(mark(x0), mark(true), x2), MARK(x0), MARK(true))
MARK(if(f(z0), x1, x2)) → c5(ACTIVE(if(active(f(mark(z0))), mark(x1), x2)), IF(mark(f(z0)), mark(x1), x2), MARK(f(z0)), MARK(x1))
MARK(if(if(z0, z1, z2), x1, x2)) → c5(ACTIVE(if(active(if(mark(z0), mark(z1), z2)), mark(x1), x2)), IF(mark(if(z0, z1, z2)), mark(x1), x2), MARK(if(z0, z1, z2)), MARK(x1))
MARK(if(c, x1, x2)) → c5(ACTIVE(if(active(c), mark(x1), x2)), IF(mark(c), mark(x1), x2), MARK(c), MARK(x1))
MARK(if(true, x1, x2)) → c5(ACTIVE(if(active(true), mark(x1), x2)), IF(mark(true), mark(x1), x2), MARK(true), MARK(x1))
MARK(if(false, x1, x2)) → c5(ACTIVE(if(active(false), mark(x1), x2)), IF(mark(false), mark(x1), x2), MARK(false), MARK(x1))
MARK(if(x0, x1, x2)) → c5(IF(mark(x0), mark(x1), x2))

K tuples:

ACTIVE(if(false, z0, z1)) → c3(MARK(z1))
MARK(f(false)) → c4(ACTIVE(f(active(false))), F(mark(false)), MARK(false))
MARK(if(x0, false, x2)) → c5(ACTIVE(if(mark(x0), active(false), x2)), IF(mark(x0), mark(false), x2), MARK(x0), MARK(false))

Defined Rule Symbols:

active, mark, f, if

Defined Pair Symbols:

ACTIVE, F, IF, MARK

Compound Symbols:

c2, c3, c9, c10, c11, c12, c13, c14, c15, c16, c1, c4, c5, c5

(19) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

MARK(if(false, x1, x2)) → c5(ACTIVE(if(active(false), mark(x1), x2)), IF(mark(false), mark(x1), x2), MARK(false), MARK(x1))

We considered the (Usable) Rules:

mark(if(z0, z1, z2)) → active(if(mark(z0), mark(z1), z2))
mark(f(z0)) → active(f(mark(z0)))
mark(c) → active(c)
mark(true) → active(true)
mark(false) → active(false)
active(f(z0)) → mark(if(z0, c, f(true)))
active(if(true, z0, z1)) → mark(z0)
active(if(false, z0, z1)) → mark(z1)
if(z0, mark(z1), z2) → if(z0, z1, z2)
if(mark(z0), 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)
f(active(z0)) → f(z0)
f(mark(z0)) → f(z0)

And the Tuples:

ACTIVE(if(true, z0, z1)) → c2(MARK(z0))
ACTIVE(if(false, z0, z1)) → c3(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))
ACTIVE(f(x0)) → c1(MARK(if(x0, c, f(true))), IF(x0, c, f(true)))
MARK(f(z0)) → c4(ACTIVE(f(z0)), F(mark(z0)), MARK(z0))
MARK(f(f(z0))) → c4(ACTIVE(f(active(f(mark(z0))))), F(mark(f(z0))), MARK(f(z0)))
MARK(f(if(z0, z1, z2))) → c4(ACTIVE(f(active(if(mark(z0), mark(z1), z2)))), F(mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
MARK(f(c)) → c4(ACTIVE(f(active(c))), F(mark(c)), MARK(c))
MARK(f(true)) → c4(ACTIVE(f(active(true))), F(mark(true)), MARK(true))
MARK(f(false)) → c4(ACTIVE(f(active(false))), F(mark(false)), MARK(false))
MARK(if(z0, x1, z2)) → c5(ACTIVE(if(z0, mark(x1), z2)), IF(mark(z0), mark(x1), z2), MARK(z0), MARK(x1))
MARK(if(x0, z1, z2)) → c5(ACTIVE(if(mark(x0), z1, z2)), IF(mark(x0), mark(z1), z2), MARK(x0), MARK(z1))
MARK(if(x0, f(z0), x2)) → c5(ACTIVE(if(mark(x0), active(f(mark(z0))), x2)), IF(mark(x0), mark(f(z0)), x2), MARK(x0), MARK(f(z0)))
MARK(if(x0, if(z0, z1, z2), x2)) → c5(ACTIVE(if(mark(x0), active(if(mark(z0), mark(z1), z2)), x2)), IF(mark(x0), mark(if(z0, z1, z2)), x2), MARK(x0), MARK(if(z0, z1, z2)))
MARK(if(x0, c, x2)) → c5(ACTIVE(if(mark(x0), active(c), x2)), IF(mark(x0), mark(c), x2), MARK(x0), MARK(c))
MARK(if(x0, true, x2)) → c5(ACTIVE(if(mark(x0), active(true), x2)), IF(mark(x0), mark(true), x2), MARK(x0), MARK(true))
MARK(if(x0, false, x2)) → c5(ACTIVE(if(mark(x0), active(false), x2)), IF(mark(x0), mark(false), x2), MARK(x0), MARK(false))
MARK(if(f(z0), x1, x2)) → c5(ACTIVE(if(active(f(mark(z0))), mark(x1), x2)), IF(mark(f(z0)), mark(x1), x2), MARK(f(z0)), MARK(x1))
MARK(if(if(z0, z1, z2), x1, x2)) → c5(ACTIVE(if(active(if(mark(z0), mark(z1), z2)), mark(x1), x2)), IF(mark(if(z0, z1, z2)), mark(x1), x2), MARK(if(z0, z1, z2)), MARK(x1))
MARK(if(c, x1, x2)) → c5(ACTIVE(if(active(c), mark(x1), x2)), IF(mark(c), mark(x1), x2), MARK(c), MARK(x1))
MARK(if(true, x1, x2)) → c5(ACTIVE(if(active(true), mark(x1), x2)), IF(mark(true), mark(x1), x2), MARK(true), MARK(x1))
MARK(if(false, x1, x2)) → c5(ACTIVE(if(active(false), mark(x1), x2)), IF(mark(false), mark(x1), x2), MARK(false), MARK(x1))
MARK(if(x0, x1, x2)) → c5(IF(mark(x0), mark(x1), x2))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVE(x₁)) = [2]x₁
POL(F(x₁)) = 0
POL(IF(x₁, x₂, x₃)) = [4]x₂ + [2]x₃
POL(MARK(x₁)) = [5]x₁
POL(active(x₁)) = x₁
POL(c) = 0
POL(c1(x₁, x₂)) = x₁ + x₂
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c4(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c5(x₁)) = x₁
POL(c5(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(c9(x₁)) = x₁
POL(f(x₁)) = [5]x₁
POL(false) = [2]
POL(if(x₁, x₂, x₃)) = [2]x₁ + [3]x₂ + [4]x₃
POL(mark(x₁)) = x₁
POL(true) = 0

(20) 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(if(true, z0, z1)) → c2(MARK(z0))
ACTIVE(if(false, z0, z1)) → c3(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))
ACTIVE(f(x0)) → c1(MARK(if(x0, c, f(true))), IF(x0, c, f(true)))
MARK(f(z0)) → c4(ACTIVE(f(z0)), F(mark(z0)), MARK(z0))
MARK(f(f(z0))) → c4(ACTIVE(f(active(f(mark(z0))))), F(mark(f(z0))), MARK(f(z0)))
MARK(f(if(z0, z1, z2))) → c4(ACTIVE(f(active(if(mark(z0), mark(z1), z2)))), F(mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
MARK(f(c)) → c4(ACTIVE(f(active(c))), F(mark(c)), MARK(c))
MARK(f(true)) → c4(ACTIVE(f(active(true))), F(mark(true)), MARK(true))
MARK(f(false)) → c4(ACTIVE(f(active(false))), F(mark(false)), MARK(false))
MARK(if(z0, x1, z2)) → c5(ACTIVE(if(z0, mark(x1), z2)), IF(mark(z0), mark(x1), z2), MARK(z0), MARK(x1))
MARK(if(x0, z1, z2)) → c5(ACTIVE(if(mark(x0), z1, z2)), IF(mark(x0), mark(z1), z2), MARK(x0), MARK(z1))
MARK(if(x0, f(z0), x2)) → c5(ACTIVE(if(mark(x0), active(f(mark(z0))), x2)), IF(mark(x0), mark(f(z0)), x2), MARK(x0), MARK(f(z0)))
MARK(if(x0, if(z0, z1, z2), x2)) → c5(ACTIVE(if(mark(x0), active(if(mark(z0), mark(z1), z2)), x2)), IF(mark(x0), mark(if(z0, z1, z2)), x2), MARK(x0), MARK(if(z0, z1, z2)))
MARK(if(x0, c, x2)) → c5(ACTIVE(if(mark(x0), active(c), x2)), IF(mark(x0), mark(c), x2), MARK(x0), MARK(c))
MARK(if(x0, true, x2)) → c5(ACTIVE(if(mark(x0), active(true), x2)), IF(mark(x0), mark(true), x2), MARK(x0), MARK(true))
MARK(if(x0, false, x2)) → c5(ACTIVE(if(mark(x0), active(false), x2)), IF(mark(x0), mark(false), x2), MARK(x0), MARK(false))
MARK(if(f(z0), x1, x2)) → c5(ACTIVE(if(active(f(mark(z0))), mark(x1), x2)), IF(mark(f(z0)), mark(x1), x2), MARK(f(z0)), MARK(x1))
MARK(if(if(z0, z1, z2), x1, x2)) → c5(ACTIVE(if(active(if(mark(z0), mark(z1), z2)), mark(x1), x2)), IF(mark(if(z0, z1, z2)), mark(x1), x2), MARK(if(z0, z1, z2)), MARK(x1))
MARK(if(c, x1, x2)) → c5(ACTIVE(if(active(c), mark(x1), x2)), IF(mark(c), mark(x1), x2), MARK(c), MARK(x1))
MARK(if(true, x1, x2)) → c5(ACTIVE(if(active(true), mark(x1), x2)), IF(mark(true), mark(x1), x2), MARK(true), MARK(x1))
MARK(if(false, x1, x2)) → c5(ACTIVE(if(active(false), mark(x1), x2)), IF(mark(false), mark(x1), x2), MARK(false), MARK(x1))
MARK(if(x0, x1, x2)) → c5(IF(mark(x0), mark(x1), x2))

S tuples:

ACTIVE(if(true, z0, z1)) → c2(MARK(z0))
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))
ACTIVE(f(x0)) → c1(MARK(if(x0, c, f(true))), IF(x0, c, f(true)))
MARK(f(z0)) → c4(ACTIVE(f(z0)), F(mark(z0)), MARK(z0))
MARK(f(f(z0))) → c4(ACTIVE(f(active(f(mark(z0))))), F(mark(f(z0))), MARK(f(z0)))
MARK(f(if(z0, z1, z2))) → c4(ACTIVE(f(active(if(mark(z0), mark(z1), z2)))), F(mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
MARK(f(c)) → c4(ACTIVE(f(active(c))), F(mark(c)), MARK(c))
MARK(f(true)) → c4(ACTIVE(f(active(true))), F(mark(true)), MARK(true))
MARK(if(z0, x1, z2)) → c5(ACTIVE(if(z0, mark(x1), z2)), IF(mark(z0), mark(x1), z2), MARK(z0), MARK(x1))
MARK(if(x0, z1, z2)) → c5(ACTIVE(if(mark(x0), z1, z2)), IF(mark(x0), mark(z1), z2), MARK(x0), MARK(z1))
MARK(if(x0, f(z0), x2)) → c5(ACTIVE(if(mark(x0), active(f(mark(z0))), x2)), IF(mark(x0), mark(f(z0)), x2), MARK(x0), MARK(f(z0)))
MARK(if(x0, if(z0, z1, z2), x2)) → c5(ACTIVE(if(mark(x0), active(if(mark(z0), mark(z1), z2)), x2)), IF(mark(x0), mark(if(z0, z1, z2)), x2), MARK(x0), MARK(if(z0, z1, z2)))
MARK(if(x0, c, x2)) → c5(ACTIVE(if(mark(x0), active(c), x2)), IF(mark(x0), mark(c), x2), MARK(x0), MARK(c))
MARK(if(x0, true, x2)) → c5(ACTIVE(if(mark(x0), active(true), x2)), IF(mark(x0), mark(true), x2), MARK(x0), MARK(true))
MARK(if(f(z0), x1, x2)) → c5(ACTIVE(if(active(f(mark(z0))), mark(x1), x2)), IF(mark(f(z0)), mark(x1), x2), MARK(f(z0)), MARK(x1))
MARK(if(if(z0, z1, z2), x1, x2)) → c5(ACTIVE(if(active(if(mark(z0), mark(z1), z2)), mark(x1), x2)), IF(mark(if(z0, z1, z2)), mark(x1), x2), MARK(if(z0, z1, z2)), MARK(x1))
MARK(if(c, x1, x2)) → c5(ACTIVE(if(active(c), mark(x1), x2)), IF(mark(c), mark(x1), x2), MARK(c), MARK(x1))
MARK(if(true, x1, x2)) → c5(ACTIVE(if(active(true), mark(x1), x2)), IF(mark(true), mark(x1), x2), MARK(true), MARK(x1))
MARK(if(x0, x1, x2)) → c5(IF(mark(x0), mark(x1), x2))

K tuples:

ACTIVE(if(false, z0, z1)) → c3(MARK(z1))
MARK(f(false)) → c4(ACTIVE(f(active(false))), F(mark(false)), MARK(false))
MARK(if(x0, false, x2)) → c5(ACTIVE(if(mark(x0), active(false), x2)), IF(mark(x0), mark(false), x2), MARK(x0), MARK(false))
MARK(if(false, x1, x2)) → c5(ACTIVE(if(active(false), mark(x1), x2)), IF(mark(false), mark(x1), x2), MARK(false), MARK(x1))

Defined Rule Symbols:

active, mark, f, if

Defined Pair Symbols:

ACTIVE, F, IF, MARK

Compound Symbols:

c2, c3, c9, c10, c11, c12, c13, c14, c15, c16, c1, c4, c5, c5

(21) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace MARK(f(f(z0))) → c4(ACTIVE(f(active(f(mark(z0))))), F(mark(f(z0))), MARK(f(z0))) by

MARK(f(f(x0))) → c4(ACTIVE(f(f(mark(x0)))), F(mark(f(x0))), MARK(f(x0)))
MARK(f(f(x0))) → c4(ACTIVE(f(mark(if(mark(x0), c, f(true))))), F(mark(f(x0))), MARK(f(x0)))
MARK(f(f(z0))) → c4(ACTIVE(f(active(f(z0)))), F(mark(f(z0))), MARK(f(z0)))
MARK(f(f(f(z0)))) → c4(ACTIVE(f(active(f(active(f(mark(z0))))))), F(mark(f(f(z0)))), MARK(f(f(z0))))
MARK(f(f(if(z0, z1, z2)))) → c4(ACTIVE(f(active(f(active(if(mark(z0), mark(z1), z2)))))), F(mark(f(if(z0, z1, z2)))), MARK(f(if(z0, z1, z2))))
MARK(f(f(c))) → c4(ACTIVE(f(active(f(active(c))))), F(mark(f(c))), MARK(f(c)))
MARK(f(f(true))) → c4(ACTIVE(f(active(f(active(true))))), F(mark(f(true))), MARK(f(true)))
MARK(f(f(false))) → c4(ACTIVE(f(active(f(active(false))))), F(mark(f(false))), MARK(f(false)))
MARK(f(f(x0))) → c4(F(mark(f(x0))))

(22) 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(if(true, z0, z1)) → c2(MARK(z0))
ACTIVE(if(false, z0, z1)) → c3(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))
ACTIVE(f(x0)) → c1(MARK(if(x0, c, f(true))), IF(x0, c, f(true)))
MARK(f(z0)) → c4(ACTIVE(f(z0)), F(mark(z0)), MARK(z0))
MARK(f(if(z0, z1, z2))) → c4(ACTIVE(f(active(if(mark(z0), mark(z1), z2)))), F(mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
MARK(f(c)) → c4(ACTIVE(f(active(c))), F(mark(c)), MARK(c))
MARK(f(true)) → c4(ACTIVE(f(active(true))), F(mark(true)), MARK(true))
MARK(f(false)) → c4(ACTIVE(f(active(false))), F(mark(false)), MARK(false))
MARK(if(z0, x1, z2)) → c5(ACTIVE(if(z0, mark(x1), z2)), IF(mark(z0), mark(x1), z2), MARK(z0), MARK(x1))
MARK(if(x0, z1, z2)) → c5(ACTIVE(if(mark(x0), z1, z2)), IF(mark(x0), mark(z1), z2), MARK(x0), MARK(z1))
MARK(if(x0, f(z0), x2)) → c5(ACTIVE(if(mark(x0), active(f(mark(z0))), x2)), IF(mark(x0), mark(f(z0)), x2), MARK(x0), MARK(f(z0)))
MARK(if(x0, if(z0, z1, z2), x2)) → c5(ACTIVE(if(mark(x0), active(if(mark(z0), mark(z1), z2)), x2)), IF(mark(x0), mark(if(z0, z1, z2)), x2), MARK(x0), MARK(if(z0, z1, z2)))
MARK(if(x0, c, x2)) → c5(ACTIVE(if(mark(x0), active(c), x2)), IF(mark(x0), mark(c), x2), MARK(x0), MARK(c))
MARK(if(x0, true, x2)) → c5(ACTIVE(if(mark(x0), active(true), x2)), IF(mark(x0), mark(true), x2), MARK(x0), MARK(true))
MARK(if(x0, false, x2)) → c5(ACTIVE(if(mark(x0), active(false), x2)), IF(mark(x0), mark(false), x2), MARK(x0), MARK(false))
MARK(if(f(z0), x1, x2)) → c5(ACTIVE(if(active(f(mark(z0))), mark(x1), x2)), IF(mark(f(z0)), mark(x1), x2), MARK(f(z0)), MARK(x1))
MARK(if(if(z0, z1, z2), x1, x2)) → c5(ACTIVE(if(active(if(mark(z0), mark(z1), z2)), mark(x1), x2)), IF(mark(if(z0, z1, z2)), mark(x1), x2), MARK(if(z0, z1, z2)), MARK(x1))
MARK(if(c, x1, x2)) → c5(ACTIVE(if(active(c), mark(x1), x2)), IF(mark(c), mark(x1), x2), MARK(c), MARK(x1))
MARK(if(true, x1, x2)) → c5(ACTIVE(if(active(true), mark(x1), x2)), IF(mark(true), mark(x1), x2), MARK(true), MARK(x1))
MARK(if(false, x1, x2)) → c5(ACTIVE(if(active(false), mark(x1), x2)), IF(mark(false), mark(x1), x2), MARK(false), MARK(x1))
MARK(if(x0, x1, x2)) → c5(IF(mark(x0), mark(x1), x2))
MARK(f(f(x0))) → c4(ACTIVE(f(f(mark(x0)))), F(mark(f(x0))), MARK(f(x0)))
MARK(f(f(x0))) → c4(ACTIVE(f(mark(if(mark(x0), c, f(true))))), F(mark(f(x0))), MARK(f(x0)))
MARK(f(f(z0))) → c4(ACTIVE(f(active(f(z0)))), F(mark(f(z0))), MARK(f(z0)))
MARK(f(f(f(z0)))) → c4(ACTIVE(f(active(f(active(f(mark(z0))))))), F(mark(f(f(z0)))), MARK(f(f(z0))))
MARK(f(f(if(z0, z1, z2)))) → c4(ACTIVE(f(active(f(active(if(mark(z0), mark(z1), z2)))))), F(mark(f(if(z0, z1, z2)))), MARK(f(if(z0, z1, z2))))
MARK(f(f(c))) → c4(ACTIVE(f(active(f(active(c))))), F(mark(f(c))), MARK(f(c)))
MARK(f(f(true))) → c4(ACTIVE(f(active(f(active(true))))), F(mark(f(true))), MARK(f(true)))
MARK(f(f(false))) → c4(ACTIVE(f(active(f(active(false))))), F(mark(f(false))), MARK(f(false)))
MARK(f(f(x0))) → c4(F(mark(f(x0))))

S tuples:

ACTIVE(if(true, z0, z1)) → c2(MARK(z0))
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))
ACTIVE(f(x0)) → c1(MARK(if(x0, c, f(true))), IF(x0, c, f(true)))
MARK(f(z0)) → c4(ACTIVE(f(z0)), F(mark(z0)), MARK(z0))
MARK(f(if(z0, z1, z2))) → c4(ACTIVE(f(active(if(mark(z0), mark(z1), z2)))), F(mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
MARK(f(c)) → c4(ACTIVE(f(active(c))), F(mark(c)), MARK(c))
MARK(f(true)) → c4(ACTIVE(f(active(true))), F(mark(true)), MARK(true))
MARK(if(z0, x1, z2)) → c5(ACTIVE(if(z0, mark(x1), z2)), IF(mark(z0), mark(x1), z2), MARK(z0), MARK(x1))
MARK(if(x0, z1, z2)) → c5(ACTIVE(if(mark(x0), z1, z2)), IF(mark(x0), mark(z1), z2), MARK(x0), MARK(z1))
MARK(if(x0, f(z0), x2)) → c5(ACTIVE(if(mark(x0), active(f(mark(z0))), x2)), IF(mark(x0), mark(f(z0)), x2), MARK(x0), MARK(f(z0)))
MARK(if(x0, if(z0, z1, z2), x2)) → c5(ACTIVE(if(mark(x0), active(if(mark(z0), mark(z1), z2)), x2)), IF(mark(x0), mark(if(z0, z1, z2)), x2), MARK(x0), MARK(if(z0, z1, z2)))
MARK(if(x0, c, x2)) → c5(ACTIVE(if(mark(x0), active(c), x2)), IF(mark(x0), mark(c), x2), MARK(x0), MARK(c))
MARK(if(x0, true, x2)) → c5(ACTIVE(if(mark(x0), active(true), x2)), IF(mark(x0), mark(true), x2), MARK(x0), MARK(true))
MARK(if(f(z0), x1, x2)) → c5(ACTIVE(if(active(f(mark(z0))), mark(x1), x2)), IF(mark(f(z0)), mark(x1), x2), MARK(f(z0)), MARK(x1))
MARK(if(if(z0, z1, z2), x1, x2)) → c5(ACTIVE(if(active(if(mark(z0), mark(z1), z2)), mark(x1), x2)), IF(mark(if(z0, z1, z2)), mark(x1), x2), MARK(if(z0, z1, z2)), MARK(x1))
MARK(if(c, x1, x2)) → c5(ACTIVE(if(active(c), mark(x1), x2)), IF(mark(c), mark(x1), x2), MARK(c), MARK(x1))
MARK(if(true, x1, x2)) → c5(ACTIVE(if(active(true), mark(x1), x2)), IF(mark(true), mark(x1), x2), MARK(true), MARK(x1))
MARK(if(x0, x1, x2)) → c5(IF(mark(x0), mark(x1), x2))
MARK(f(f(x0))) → c4(ACTIVE(f(f(mark(x0)))), F(mark(f(x0))), MARK(f(x0)))
MARK(f(f(x0))) → c4(ACTIVE(f(mark(if(mark(x0), c, f(true))))), F(mark(f(x0))), MARK(f(x0)))
MARK(f(f(z0))) → c4(ACTIVE(f(active(f(z0)))), F(mark(f(z0))), MARK(f(z0)))
MARK(f(f(f(z0)))) → c4(ACTIVE(f(active(f(active(f(mark(z0))))))), F(mark(f(f(z0)))), MARK(f(f(z0))))
MARK(f(f(if(z0, z1, z2)))) → c4(ACTIVE(f(active(f(active(if(mark(z0), mark(z1), z2)))))), F(mark(f(if(z0, z1, z2)))), MARK(f(if(z0, z1, z2))))
MARK(f(f(c))) → c4(ACTIVE(f(active(f(active(c))))), F(mark(f(c))), MARK(f(c)))
MARK(f(f(true))) → c4(ACTIVE(f(active(f(active(true))))), F(mark(f(true))), MARK(f(true)))
MARK(f(f(false))) → c4(ACTIVE(f(active(f(active(false))))), F(mark(f(false))), MARK(f(false)))
MARK(f(f(x0))) → c4(F(mark(f(x0))))

K tuples:

ACTIVE(if(false, z0, z1)) → c3(MARK(z1))
MARK(f(false)) → c4(ACTIVE(f(active(false))), F(mark(false)), MARK(false))
MARK(if(x0, false, x2)) → c5(ACTIVE(if(mark(x0), active(false), x2)), IF(mark(x0), mark(false), x2), MARK(x0), MARK(false))
MARK(if(false, x1, x2)) → c5(ACTIVE(if(active(false), mark(x1), x2)), IF(mark(false), mark(x1), x2), MARK(false), MARK(x1))

Defined Rule Symbols:

active, mark, f, if

Defined Pair Symbols:

ACTIVE, F, IF, MARK

Compound Symbols:

c2, c3, c9, c10, c11, c12, c13, c14, c15, c16, c1, c4, c5, c5, c4

(23) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

MARK(f(f(false))) → c4(ACTIVE(f(active(f(active(false))))), F(mark(f(false))), MARK(f(false)))

We considered the (Usable) Rules:

mark(if(z0, z1, z2)) → active(if(mark(z0), mark(z1), z2))
mark(f(z0)) → active(f(mark(z0)))
mark(c) → active(c)
mark(true) → active(true)
mark(false) → active(false)
active(f(z0)) → mark(if(z0, c, f(true)))
active(if(true, z0, z1)) → mark(z0)
active(if(false, z0, z1)) → mark(z1)
if(z0, mark(z1), z2) → if(z0, z1, z2)
if(mark(z0), 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)
f(active(z0)) → f(z0)
f(mark(z0)) → f(z0)

And the Tuples:

ACTIVE(if(true, z0, z1)) → c2(MARK(z0))
ACTIVE(if(false, z0, z1)) → c3(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))
ACTIVE(f(x0)) → c1(MARK(if(x0, c, f(true))), IF(x0, c, f(true)))
MARK(f(z0)) → c4(ACTIVE(f(z0)), F(mark(z0)), MARK(z0))
MARK(f(if(z0, z1, z2))) → c4(ACTIVE(f(active(if(mark(z0), mark(z1), z2)))), F(mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
MARK(f(c)) → c4(ACTIVE(f(active(c))), F(mark(c)), MARK(c))
MARK(f(true)) → c4(ACTIVE(f(active(true))), F(mark(true)), MARK(true))
MARK(f(false)) → c4(ACTIVE(f(active(false))), F(mark(false)), MARK(false))
MARK(if(z0, x1, z2)) → c5(ACTIVE(if(z0, mark(x1), z2)), IF(mark(z0), mark(x1), z2), MARK(z0), MARK(x1))
MARK(if(x0, z1, z2)) → c5(ACTIVE(if(mark(x0), z1, z2)), IF(mark(x0), mark(z1), z2), MARK(x0), MARK(z1))
MARK(if(x0, f(z0), x2)) → c5(ACTIVE(if(mark(x0), active(f(mark(z0))), x2)), IF(mark(x0), mark(f(z0)), x2), MARK(x0), MARK(f(z0)))
MARK(if(x0, if(z0, z1, z2), x2)) → c5(ACTIVE(if(mark(x0), active(if(mark(z0), mark(z1), z2)), x2)), IF(mark(x0), mark(if(z0, z1, z2)), x2), MARK(x0), MARK(if(z0, z1, z2)))
MARK(if(x0, c, x2)) → c5(ACTIVE(if(mark(x0), active(c), x2)), IF(mark(x0), mark(c), x2), MARK(x0), MARK(c))
MARK(if(x0, true, x2)) → c5(ACTIVE(if(mark(x0), active(true), x2)), IF(mark(x0), mark(true), x2), MARK(x0), MARK(true))
MARK(if(x0, false, x2)) → c5(ACTIVE(if(mark(x0), active(false), x2)), IF(mark(x0), mark(false), x2), MARK(x0), MARK(false))
MARK(if(f(z0), x1, x2)) → c5(ACTIVE(if(active(f(mark(z0))), mark(x1), x2)), IF(mark(f(z0)), mark(x1), x2), MARK(f(z0)), MARK(x1))
MARK(if(if(z0, z1, z2), x1, x2)) → c5(ACTIVE(if(active(if(mark(z0), mark(z1), z2)), mark(x1), x2)), IF(mark(if(z0, z1, z2)), mark(x1), x2), MARK(if(z0, z1, z2)), MARK(x1))
MARK(if(c, x1, x2)) → c5(ACTIVE(if(active(c), mark(x1), x2)), IF(mark(c), mark(x1), x2), MARK(c), MARK(x1))
MARK(if(true, x1, x2)) → c5(ACTIVE(if(active(true), mark(x1), x2)), IF(mark(true), mark(x1), x2), MARK(true), MARK(x1))
MARK(if(false, x1, x2)) → c5(ACTIVE(if(active(false), mark(x1), x2)), IF(mark(false), mark(x1), x2), MARK(false), MARK(x1))
MARK(if(x0, x1, x2)) → c5(IF(mark(x0), mark(x1), x2))
MARK(f(f(x0))) → c4(ACTIVE(f(f(mark(x0)))), F(mark(f(x0))), MARK(f(x0)))
MARK(f(f(x0))) → c4(ACTIVE(f(mark(if(mark(x0), c, f(true))))), F(mark(f(x0))), MARK(f(x0)))
MARK(f(f(z0))) → c4(ACTIVE(f(active(f(z0)))), F(mark(f(z0))), MARK(f(z0)))
MARK(f(f(f(z0)))) → c4(ACTIVE(f(active(f(active(f(mark(z0))))))), F(mark(f(f(z0)))), MARK(f(f(z0))))
MARK(f(f(if(z0, z1, z2)))) → c4(ACTIVE(f(active(f(active(if(mark(z0), mark(z1), z2)))))), F(mark(f(if(z0, z1, z2)))), MARK(f(if(z0, z1, z2))))
MARK(f(f(c))) → c4(ACTIVE(f(active(f(active(c))))), F(mark(f(c))), MARK(f(c)))
MARK(f(f(true))) → c4(ACTIVE(f(active(f(active(true))))), F(mark(f(true))), MARK(f(true)))
MARK(f(f(false))) → c4(ACTIVE(f(active(f(active(false))))), F(mark(f(false))), MARK(f(false)))
MARK(f(f(x0))) → c4(F(mark(f(x0))))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVE(x₁)) = x₁
POL(F(x₁)) = 0
POL(IF(x₁, x₂, x₃)) = 0
POL(MARK(x₁)) = [2]x₁
POL(active(x₁)) = x₁
POL(c) = 0
POL(c1(x₁, x₂)) = x₁ + x₂
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c4(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c5(x₁)) = x₁
POL(c5(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(c9(x₁)) = x₁
POL(f(x₁)) = [4]x₁
POL(false) = [1]
POL(if(x₁, x₂, x₃)) = [2]x₁ + [4]x₂ + [4]x₃
POL(mark(x₁)) = x₁
POL(true) = 0

(24) 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(if(true, z0, z1)) → c2(MARK(z0))
ACTIVE(if(false, z0, z1)) → c3(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))
ACTIVE(f(x0)) → c1(MARK(if(x0, c, f(true))), IF(x0, c, f(true)))
MARK(f(z0)) → c4(ACTIVE(f(z0)), F(mark(z0)), MARK(z0))
MARK(f(if(z0, z1, z2))) → c4(ACTIVE(f(active(if(mark(z0), mark(z1), z2)))), F(mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
MARK(f(c)) → c4(ACTIVE(f(active(c))), F(mark(c)), MARK(c))
MARK(f(true)) → c4(ACTIVE(f(active(true))), F(mark(true)), MARK(true))
MARK(f(false)) → c4(ACTIVE(f(active(false))), F(mark(false)), MARK(false))
MARK(if(z0, x1, z2)) → c5(ACTIVE(if(z0, mark(x1), z2)), IF(mark(z0), mark(x1), z2), MARK(z0), MARK(x1))
MARK(if(x0, z1, z2)) → c5(ACTIVE(if(mark(x0), z1, z2)), IF(mark(x0), mark(z1), z2), MARK(x0), MARK(z1))
MARK(if(x0, f(z0), x2)) → c5(ACTIVE(if(mark(x0), active(f(mark(z0))), x2)), IF(mark(x0), mark(f(z0)), x2), MARK(x0), MARK(f(z0)))
MARK(if(x0, if(z0, z1, z2), x2)) → c5(ACTIVE(if(mark(x0), active(if(mark(z0), mark(z1), z2)), x2)), IF(mark(x0), mark(if(z0, z1, z2)), x2), MARK(x0), MARK(if(z0, z1, z2)))
MARK(if(x0, c, x2)) → c5(ACTIVE(if(mark(x0), active(c), x2)), IF(mark(x0), mark(c), x2), MARK(x0), MARK(c))
MARK(if(x0, true, x2)) → c5(ACTIVE(if(mark(x0), active(true), x2)), IF(mark(x0), mark(true), x2), MARK(x0), MARK(true))
MARK(if(x0, false, x2)) → c5(ACTIVE(if(mark(x0), active(false), x2)), IF(mark(x0), mark(false), x2), MARK(x0), MARK(false))
MARK(if(f(z0), x1, x2)) → c5(ACTIVE(if(active(f(mark(z0))), mark(x1), x2)), IF(mark(f(z0)), mark(x1), x2), MARK(f(z0)), MARK(x1))
MARK(if(if(z0, z1, z2), x1, x2)) → c5(ACTIVE(if(active(if(mark(z0), mark(z1), z2)), mark(x1), x2)), IF(mark(if(z0, z1, z2)), mark(x1), x2), MARK(if(z0, z1, z2)), MARK(x1))
MARK(if(c, x1, x2)) → c5(ACTIVE(if(active(c), mark(x1), x2)), IF(mark(c), mark(x1), x2), MARK(c), MARK(x1))
MARK(if(true, x1, x2)) → c5(ACTIVE(if(active(true), mark(x1), x2)), IF(mark(true), mark(x1), x2), MARK(true), MARK(x1))
MARK(if(false, x1, x2)) → c5(ACTIVE(if(active(false), mark(x1), x2)), IF(mark(false), mark(x1), x2), MARK(false), MARK(x1))
MARK(if(x0, x1, x2)) → c5(IF(mark(x0), mark(x1), x2))
MARK(f(f(x0))) → c4(ACTIVE(f(f(mark(x0)))), F(mark(f(x0))), MARK(f(x0)))
MARK(f(f(x0))) → c4(ACTIVE(f(mark(if(mark(x0), c, f(true))))), F(mark(f(x0))), MARK(f(x0)))
MARK(f(f(z0))) → c4(ACTIVE(f(active(f(z0)))), F(mark(f(z0))), MARK(f(z0)))
MARK(f(f(f(z0)))) → c4(ACTIVE(f(active(f(active(f(mark(z0))))))), F(mark(f(f(z0)))), MARK(f(f(z0))))
MARK(f(f(if(z0, z1, z2)))) → c4(ACTIVE(f(active(f(active(if(mark(z0), mark(z1), z2)))))), F(mark(f(if(z0, z1, z2)))), MARK(f(if(z0, z1, z2))))
MARK(f(f(c))) → c4(ACTIVE(f(active(f(active(c))))), F(mark(f(c))), MARK(f(c)))
MARK(f(f(true))) → c4(ACTIVE(f(active(f(active(true))))), F(mark(f(true))), MARK(f(true)))
MARK(f(f(false))) → c4(ACTIVE(f(active(f(active(false))))), F(mark(f(false))), MARK(f(false)))
MARK(f(f(x0))) → c4(F(mark(f(x0))))

S tuples:

ACTIVE(if(true, z0, z1)) → c2(MARK(z0))
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))
ACTIVE(f(x0)) → c1(MARK(if(x0, c, f(true))), IF(x0, c, f(true)))
MARK(f(z0)) → c4(ACTIVE(f(z0)), F(mark(z0)), MARK(z0))
MARK(f(if(z0, z1, z2))) → c4(ACTIVE(f(active(if(mark(z0), mark(z1), z2)))), F(mark(if(z0, z1, z2))), MARK(if(z0, z1, z2)))
MARK(f(c)) → c4(ACTIVE(f(active(c))), F(mark(c)), MARK(c))
MARK(f(true)) → c4(ACTIVE(f(active(true))), F(mark(true)), MARK(true))
MARK(if(z0, x1, z2)) → c5(ACTIVE(if(z0, mark(x1), z2)), IF(mark(z0), mark(x1), z2), MARK(z0), MARK(x1))
MARK(if(x0, z1, z2)) → c5(ACTIVE(if(mark(x0), z1, z2)), IF(mark(x0), mark(z1), z2), MARK(x0), MARK(z1))
MARK(if(x0, f(z0), x2)) → c5(ACTIVE(if(mark(x0), active(f(mark(z0))), x2)), IF(mark(x0), mark(f(z0)), x2), MARK(x0), MARK(f(z0)))
MARK(if(x0, if(z0, z1, z2), x2)) → c5(ACTIVE(if(mark(x0), active(if(mark(z0), mark(z1), z2)), x2)), IF(mark(x0), mark(if(z0, z1, z2)), x2), MARK(x0), MARK(if(z0, z1, z2)))
MARK(if(x0, c, x2)) → c5(ACTIVE(if(mark(x0), active(c), x2)), IF(mark(x0), mark(c), x2), MARK(x0), MARK(c))
MARK(if(x0, true, x2)) → c5(ACTIVE(if(mark(x0), active(true), x2)), IF(mark(x0), mark(true), x2), MARK(x0), MARK(true))
MARK(if(f(z0), x1, x2)) → c5(ACTIVE(if(active(f(mark(z0))), mark(x1), x2)), IF(mark(f(z0)), mark(x1), x2), MARK(f(z0)), MARK(x1))
MARK(if(if(z0, z1, z2), x1, x2)) → c5(ACTIVE(if(active(if(mark(z0), mark(z1), z2)), mark(x1), x2)), IF(mark(if(z0, z1, z2)), mark(x1), x2), MARK(if(z0, z1, z2)), MARK(x1))
MARK(if(c, x1, x2)) → c5(ACTIVE(if(active(c), mark(x1), x2)), IF(mark(c), mark(x1), x2), MARK(c), MARK(x1))
MARK(if(true, x1, x2)) → c5(ACTIVE(if(active(true), mark(x1), x2)), IF(mark(true), mark(x1), x2), MARK(true), MARK(x1))
MARK(if(x0, x1, x2)) → c5(IF(mark(x0), mark(x1), x2))
MARK(f(f(x0))) → c4(ACTIVE(f(f(mark(x0)))), F(mark(f(x0))), MARK(f(x0)))
MARK(f(f(x0))) → c4(ACTIVE(f(mark(if(mark(x0), c, f(true))))), F(mark(f(x0))), MARK(f(x0)))
MARK(f(f(z0))) → c4(ACTIVE(f(active(f(z0)))), F(mark(f(z0))), MARK(f(z0)))
MARK(f(f(f(z0)))) → c4(ACTIVE(f(active(f(active(f(mark(z0))))))), F(mark(f(f(z0)))), MARK(f(f(z0))))
MARK(f(f(if(z0, z1, z2)))) → c4(ACTIVE(f(active(f(active(if(mark(z0), mark(z1), z2)))))), F(mark(f(if(z0, z1, z2)))), MARK(f(if(z0, z1, z2))))
MARK(f(f(c))) → c4(ACTIVE(f(active(f(active(c))))), F(mark(f(c))), MARK(f(c)))
MARK(f(f(true))) → c4(ACTIVE(f(active(f(active(true))))), F(mark(f(true))), MARK(f(true)))
MARK(f(f(x0))) → c4(F(mark(f(x0))))

K tuples:

ACTIVE(if(false, z0, z1)) → c3(MARK(z1))
MARK(f(false)) → c4(ACTIVE(f(active(false))), F(mark(false)), MARK(false))
MARK(if(x0, false, x2)) → c5(ACTIVE(if(mark(x0), active(false), x2)), IF(mark(x0), mark(false), x2), MARK(x0), MARK(false))
MARK(if(false, x1, x2)) → c5(ACTIVE(if(active(false), mark(x1), x2)), IF(mark(false), mark(x1), x2), MARK(false), MARK(x1))
MARK(f(f(false))) → c4(ACTIVE(f(active(f(active(false))))), F(mark(f(false))), MARK(f(false)))

Defined Rule Symbols:

active, mark, f, if

Defined Pair Symbols:

ACTIVE, F, IF, MARK

Compound Symbols:

c2, c3, c9, c10, c11, c12, c13, c14, c15, c16, c1, c4, c5, c5, c4

(25) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace MARK(f(if(z0, z1, z2))) → c4(ACTIVE(f(active(if(mark(z0), mark(z1), z2)))), F(mark(if(z0, z1, z2))), MARK(if(z0, z1, z2))) by

MARK(f(if(x0, x1, x2))) → c4(ACTIVE(f(if(mark(x0), mark(x1), x2))), F(mark(if(x0, x1, x2))), MARK(if(x0, x1, x2)))
MARK(f(if(z0, x1, z2))) → c4(ACTIVE(f(active(if(z0, mark(x1), z2)))), F(mark(if(z0, x1, z2))), MARK(if(z0, x1, z2)))
MARK(f(if(x0, z1, z2))) → c4(ACTIVE(f(active(if(mark(x0), z1, z2)))), F(mark(if(x0, z1, z2))), MARK(if(x0, z1, z2)))
MARK(f(if(x0, f(z0), x2))) → c4(ACTIVE(f(active(if(mark(x0), active(f(mark(z0))), x2)))), F(mark(if(x0, f(z0), x2))), MARK(if(x0, f(z0), x2)))
MARK(f(if(x0, if(z0, z1, z2), x2))) → c4(ACTIVE(f(active(if(mark(x0), active(if(mark(z0), mark(z1), z2)), x2)))), F(mark(if(x0, if(z0, z1, z2), x2))), MARK(if(x0, if(z0, z1, z2), x2)))
MARK(f(if(x0, c, x2))) → c4(ACTIVE(f(active(if(mark(x0), active(c), x2)))), F(mark(if(x0, c, x2))), MARK(if(x0, c, x2)))
MARK(f(if(x0, true, x2))) → c4(ACTIVE(f(active(if(mark(x0), active(true), x2)))), F(mark(if(x0, true, x2))), MARK(if(x0, true, x2)))
MARK(f(if(x0, false, x2))) → c4(ACTIVE(f(active(if(mark(x0), active(false), x2)))), F(mark(if(x0, false, x2))), MARK(if(x0, false, x2)))
MARK(f(if(f(z0), x1, x2))) → c4(ACTIVE(f(active(if(active(f(mark(z0))), mark(x1), x2)))), F(mark(if(f(z0), x1, x2))), MARK(if(f(z0), x1, x2)))
MARK(f(if(if(z0, z1, z2), x1, x2))) → c4(ACTIVE(f(active(if(active(if(mark(z0), mark(z1), z2)), mark(x1), x2)))), F(mark(if(if(z0, z1, z2), x1, x2))), MARK(if(if(z0, z1, z2), x1, x2)))
MARK(f(if(c, x1, x2))) → c4(ACTIVE(f(active(if(active(c), mark(x1), x2)))), F(mark(if(c, x1, x2))), MARK(if(c, x1, x2)))
MARK(f(if(true, x1, x2))) → c4(ACTIVE(f(active(if(active(true), mark(x1), x2)))), F(mark(if(true, x1, x2))), MARK(if(true, x1, x2)))
MARK(f(if(false, x1, x2))) → c4(ACTIVE(f(active(if(active(false), mark(x1), x2)))), F(mark(if(false, x1, x2))), MARK(if(false, x1, x2)))
MARK(f(if(x0, x1, x2))) → c4(F(mark(if(x0, x1, x2))), MARK(if(x0, x1, x2)))

(26) 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(if(true, z0, z1)) → c2(MARK(z0))
ACTIVE(if(false, z0, z1)) → c3(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))
ACTIVE(f(x0)) → c1(MARK(if(x0, c, f(true))), IF(x0, c, f(true)))
MARK(f(z0)) → c4(ACTIVE(f(z0)), F(mark(z0)), MARK(z0))
MARK(f(c)) → c4(ACTIVE(f(active(c))), F(mark(c)), MARK(c))
MARK(f(true)) → c4(ACTIVE(f(active(true))), F(mark(true)), MARK(true))
MARK(f(false)) → c4(ACTIVE(f(active(false))), F(mark(false)), MARK(false))
MARK(if(z0, x1, z2)) → c5(ACTIVE(if(z0, mark(x1), z2)), IF(mark(z0), mark(x1), z2), MARK(z0), MARK(x1))
MARK(if(x0, z1, z2)) → c5(ACTIVE(if(mark(x0), z1, z2)), IF(mark(x0), mark(z1), z2), MARK(x0), MARK(z1))
MARK(if(x0, f(z0), x2)) → c5(ACTIVE(if(mark(x0), active(f(mark(z0))), x2)), IF(mark(x0), mark(f(z0)), x2), MARK(x0), MARK(f(z0)))
MARK(if(x0, if(z0, z1, z2), x2)) → c5(ACTIVE(if(mark(x0), active(if(mark(z0), mark(z1), z2)), x2)), IF(mark(x0), mark(if(z0, z1, z2)), x2), MARK(x0), MARK(if(z0, z1, z2)))
MARK(if(x0, c, x2)) → c5(ACTIVE(if(mark(x0), active(c), x2)), IF(mark(x0), mark(c), x2), MARK(x0), MARK(c))
MARK(if(x0, true, x2)) → c5(ACTIVE(if(mark(x0), active(true), x2)), IF(mark(x0), mark(true), x2), MARK(x0), MARK(true))
MARK(if(x0, false, x2)) → c5(ACTIVE(if(mark(x0), active(false), x2)), IF(mark(x0), mark(false), x2), MARK(x0), MARK(false))
MARK(if(f(z0), x1, x2)) → c5(ACTIVE(if(active(f(mark(z0))), mark(x1), x2)), IF(mark(f(z0)), mark(x1), x2), MARK(f(z0)), MARK(x1))
MARK(if(if(z0, z1, z2), x1, x2)) → c5(ACTIVE(if(active(if(mark(z0), mark(z1), z2)), mark(x1), x2)), IF(mark(if(z0, z1, z2)), mark(x1), x2), MARK(if(z0, z1, z2)), MARK(x1))
MARK(if(c, x1, x2)) → c5(ACTIVE(if(active(c), mark(x1), x2)), IF(mark(c), mark(x1), x2), MARK(c), MARK(x1))
MARK(if(true, x1, x2)) → c5(ACTIVE(if(active(true), mark(x1), x2)), IF(mark(true), mark(x1), x2), MARK(true), MARK(x1))
MARK(if(false, x1, x2)) → c5(ACTIVE(if(active(false), mark(x1), x2)), IF(mark(false), mark(x1), x2), MARK(false), MARK(x1))
MARK(if(x0, x1, x2)) → c5(IF(mark(x0), mark(x1), x2))
MARK(f(f(x0))) → c4(ACTIVE(f(f(mark(x0)))), F(mark(f(x0))), MARK(f(x0)))
MARK(f(f(x0))) → c4(ACTIVE(f(mark(if(mark(x0), c, f(true))))), F(mark(f(x0))), MARK(f(x0)))
MARK(f(f(z0))) → c4(ACTIVE(f(active(f(z0)))), F(mark(f(z0))), MARK(f(z0)))
MARK(f(f(f(z0)))) → c4(ACTIVE(f(active(f(active(f(mark(z0))))))), F(mark(f(f(z0)))), MARK(f(f(z0))))
MARK(f(f(if(z0, z1, z2)))) → c4(ACTIVE(f(active(f(active(if(mark(z0), mark(z1), z2)))))), F(mark(f(if(z0, z1, z2)))), MARK(f(if(z0, z1, z2))))
MARK(f(f(c))) → c4(ACTIVE(f(active(f(active(c))))), F(mark(f(c))), MARK(f(c)))
MARK(f(f(true))) → c4(ACTIVE(f(active(f(active(true))))), F(mark(f(true))), MARK(f(true)))
MARK(f(f(false))) → c4(ACTIVE(f(active(f(active(false))))), F(mark(f(false))), MARK(f(false)))
MARK(f(f(x0))) → c4(F(mark(f(x0))))
MARK(f(if(x0, x1, x2))) → c4(ACTIVE(f(if(mark(x0), mark(x1), x2))), F(mark(if(x0, x1, x2))), MARK(if(x0, x1, x2)))
MARK(f(if(z0, x1, z2))) → c4(ACTIVE(f(active(if(z0, mark(x1), z2)))), F(mark(if(z0, x1, z2))), MARK(if(z0, x1, z2)))
MARK(f(if(x0, z1, z2))) → c4(ACTIVE(f(active(if(mark(x0), z1, z2)))), F(mark(if(x0, z1, z2))), MARK(if(x0, z1, z2)))
MARK(f(if(x0, f(z0), x2))) → c4(ACTIVE(f(active(if(mark(x0), active(f(mark(z0))), x2)))), F(mark(if(x0, f(z0), x2))), MARK(if(x0, f(z0), x2)))
MARK(f(if(x0, if(z0, z1, z2), x2))) → c4(ACTIVE(f(active(if(mark(x0), active(if(mark(z0), mark(z1), z2)), x2)))), F(mark(if(x0, if(z0, z1, z2), x2))), MARK(if(x0, if(z0, z1, z2), x2)))
MARK(f(if(x0, c, x2))) → c4(ACTIVE(f(active(if(mark(x0), active(c), x2)))), F(mark(if(x0, c, x2))), MARK(if(x0, c, x2)))
MARK(f(if(x0, true, x2))) → c4(ACTIVE(f(active(if(mark(x0), active(true), x2)))), F(mark(if(x0, true, x2))), MARK(if(x0, true, x2)))
MARK(f(if(x0, false, x2))) → c4(ACTIVE(f(active(if(mark(x0), active(false), x2)))), F(mark(if(x0, false, x2))), MARK(if(x0, false, x2)))
MARK(f(if(f(z0), x1, x2))) → c4(ACTIVE(f(active(if(active(f(mark(z0))), mark(x1), x2)))), F(mark(if(f(z0), x1, x2))), MARK(if(f(z0), x1, x2)))
MARK(f(if(if(z0, z1, z2), x1, x2))) → c4(ACTIVE(f(active(if(active(if(mark(z0), mark(z1), z2)), mark(x1), x2)))), F(mark(if(if(z0, z1, z2), x1, x2))), MARK(if(if(z0, z1, z2), x1, x2)))
MARK(f(if(c, x1, x2))) → c4(ACTIVE(f(active(if(active(c), mark(x1), x2)))), F(mark(if(c, x1, x2))), MARK(if(c, x1, x2)))
MARK(f(if(true, x1, x2))) → c4(ACTIVE(f(active(if(active(true), mark(x1), x2)))), F(mark(if(true, x1, x2))), MARK(if(true, x1, x2)))
MARK(f(if(false, x1, x2))) → c4(ACTIVE(f(active(if(active(false), mark(x1), x2)))), F(mark(if(false, x1, x2))), MARK(if(false, x1, x2)))
MARK(f(if(x0, x1, x2))) → c4(F(mark(if(x0, x1, x2))), MARK(if(x0, x1, x2)))

S tuples:

ACTIVE(if(true, z0, z1)) → c2(MARK(z0))
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))
ACTIVE(f(x0)) → c1(MARK(if(x0, c, f(true))), IF(x0, c, f(true)))
MARK(f(z0)) → c4(ACTIVE(f(z0)), F(mark(z0)), MARK(z0))
MARK(f(c)) → c4(ACTIVE(f(active(c))), F(mark(c)), MARK(c))
MARK(f(true)) → c4(ACTIVE(f(active(true))), F(mark(true)), MARK(true))
MARK(if(z0, x1, z2)) → c5(ACTIVE(if(z0, mark(x1), z2)), IF(mark(z0), mark(x1), z2), MARK(z0), MARK(x1))
MARK(if(x0, z1, z2)) → c5(ACTIVE(if(mark(x0), z1, z2)), IF(mark(x0), mark(z1), z2), MARK(x0), MARK(z1))
MARK(if(x0, f(z0), x2)) → c5(ACTIVE(if(mark(x0), active(f(mark(z0))), x2)), IF(mark(x0), mark(f(z0)), x2), MARK(x0), MARK(f(z0)))
MARK(if(x0, if(z0, z1, z2), x2)) → c5(ACTIVE(if(mark(x0), active(if(mark(z0), mark(z1), z2)), x2)), IF(mark(x0), mark(if(z0, z1, z2)), x2), MARK(x0), MARK(if(z0, z1, z2)))
MARK(if(x0, c, x2)) → c5(ACTIVE(if(mark(x0), active(c), x2)), IF(mark(x0), mark(c), x2), MARK(x0), MARK(c))
MARK(if(x0, true, x2)) → c5(ACTIVE(if(mark(x0), active(true), x2)), IF(mark(x0), mark(true), x2), MARK(x0), MARK(true))
MARK(if(f(z0), x1, x2)) → c5(ACTIVE(if(active(f(mark(z0))), mark(x1), x2)), IF(mark(f(z0)), mark(x1), x2), MARK(f(z0)), MARK(x1))
MARK(if(if(z0, z1, z2), x1, x2)) → c5(ACTIVE(if(active(if(mark(z0), mark(z1), z2)), mark(x1), x2)), IF(mark(if(z0, z1, z2)), mark(x1), x2), MARK(if(z0, z1, z2)), MARK(x1))
MARK(if(c, x1, x2)) → c5(ACTIVE(if(active(c), mark(x1), x2)), IF(mark(c), mark(x1), x2), MARK(c), MARK(x1))
MARK(if(true, x1, x2)) → c5(ACTIVE(if(active(true), mark(x1), x2)), IF(mark(true), mark(x1), x2), MARK(true), MARK(x1))
MARK(if(x0, x1, x2)) → c5(IF(mark(x0), mark(x1), x2))
MARK(f(f(x0))) → c4(ACTIVE(f(f(mark(x0)))), F(mark(f(x0))), MARK(f(x0)))
MARK(f(f(x0))) → c4(ACTIVE(f(mark(if(mark(x0), c, f(true))))), F(mark(f(x0))), MARK(f(x0)))
MARK(f(f(z0))) → c4(ACTIVE(f(active(f(z0)))), F(mark(f(z0))), MARK(f(z0)))
MARK(f(f(f(z0)))) → c4(ACTIVE(f(active(f(active(f(mark(z0))))))), F(mark(f(f(z0)))), MARK(f(f(z0))))
MARK(f(f(if(z0, z1, z2)))) → c4(ACTIVE(f(active(f(active(if(mark(z0), mark(z1), z2)))))), F(mark(f(if(z0, z1, z2)))), MARK(f(if(z0, z1, z2))))
MARK(f(f(c))) → c4(ACTIVE(f(active(f(active(c))))), F(mark(f(c))), MARK(f(c)))
MARK(f(f(true))) → c4(ACTIVE(f(active(f(active(true))))), F(mark(f(true))), MARK(f(true)))
MARK(f(f(x0))) → c4(F(mark(f(x0))))
MARK(f(if(x0, x1, x2))) → c4(ACTIVE(f(if(mark(x0), mark(x1), x2))), F(mark(if(x0, x1, x2))), MARK(if(x0, x1, x2)))
MARK(f(if(z0, x1, z2))) → c4(ACTIVE(f(active(if(z0, mark(x1), z2)))), F(mark(if(z0, x1, z2))), MARK(if(z0, x1, z2)))
MARK(f(if(x0, z1, z2))) → c4(ACTIVE(f(active(if(mark(x0), z1, z2)))), F(mark(if(x0, z1, z2))), MARK(if(x0, z1, z2)))
MARK(f(if(x0, f(z0), x2))) → c4(ACTIVE(f(active(if(mark(x0), active(f(mark(z0))), x2)))), F(mark(if(x0, f(z0), x2))), MARK(if(x0, f(z0), x2)))
MARK(f(if(x0, if(z0, z1, z2), x2))) → c4(ACTIVE(f(active(if(mark(x0), active(if(mark(z0), mark(z1), z2)), x2)))), F(mark(if(x0, if(z0, z1, z2), x2))), MARK(if(x0, if(z0, z1, z2), x2)))
MARK(f(if(x0, c, x2))) → c4(ACTIVE(f(active(if(mark(x0), active(c), x2)))), F(mark(if(x0, c, x2))), MARK(if(x0, c, x2)))
MARK(f(if(x0, true, x2))) → c4(ACTIVE(f(active(if(mark(x0), active(true), x2)))), F(mark(if(x0, true, x2))), MARK(if(x0, true, x2)))
MARK(f(if(x0, false, x2))) → c4(ACTIVE(f(active(if(mark(x0), active(false), x2)))), F(mark(if(x0, false, x2))), MARK(if(x0, false, x2)))
MARK(f(if(f(z0), x1, x2))) → c4(ACTIVE(f(active(if(active(f(mark(z0))), mark(x1), x2)))), F(mark(if(f(z0), x1, x2))), MARK(if(f(z0), x1, x2)))
MARK(f(if(if(z0, z1, z2), x1, x2))) → c4(ACTIVE(f(active(if(active(if(mark(z0), mark(z1), z2)), mark(x1), x2)))), F(mark(if(if(z0, z1, z2), x1, x2))), MARK(if(if(z0, z1, z2), x1, x2)))
MARK(f(if(c, x1, x2))) → c4(ACTIVE(f(active(if(active(c), mark(x1), x2)))), F(mark(if(c, x1, x2))), MARK(if(c, x1, x2)))
MARK(f(if(true, x1, x2))) → c4(ACTIVE(f(active(if(active(true), mark(x1), x2)))), F(mark(if(true, x1, x2))), MARK(if(true, x1, x2)))
MARK(f(if(false, x1, x2))) → c4(ACTIVE(f(active(if(active(false), mark(x1), x2)))), F(mark(if(false, x1, x2))), MARK(if(false, x1, x2)))
MARK(f(if(x0, x1, x2))) → c4(F(mark(if(x0, x1, x2))), MARK(if(x0, x1, x2)))

K tuples:

ACTIVE(if(false, z0, z1)) → c3(MARK(z1))
MARK(f(false)) → c4(ACTIVE(f(active(false))), F(mark(false)), MARK(false))
MARK(if(x0, false, x2)) → c5(ACTIVE(if(mark(x0), active(false), x2)), IF(mark(x0), mark(false), x2), MARK(x0), MARK(false))
MARK(if(false, x1, x2)) → c5(ACTIVE(if(active(false), mark(x1), x2)), IF(mark(false), mark(x1), x2), MARK(false), MARK(x1))
MARK(f(f(false))) → c4(ACTIVE(f(active(f(active(false))))), F(mark(f(false))), MARK(f(false)))

Defined Rule Symbols:

active, mark, f, if

Defined Pair Symbols:

ACTIVE, F, IF, MARK

Compound Symbols:

c2, c3, c9, c10, c11, c12, c13, c14, c15, c16, c1, c4, c5, c5, c4, c4

(27) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

MARK(f(if(x0, false, x2))) → c4(ACTIVE(f(active(if(mark(x0), active(false), x2)))), F(mark(if(x0, false, x2))), MARK(if(x0, false, x2)))
MARK(f(if(false, x1, x2))) → c4(ACTIVE(f(active(if(active(false), mark(x1), x2)))), F(mark(if(false, x1, x2))), MARK(if(false, x1, x2)))

We considered the (Usable) Rules:

mark(if(z0, z1, z2)) → active(if(mark(z0), mark(z1), z2))
mark(f(z0)) → active(f(mark(z0)))
mark(c) → active(c)
mark(true) → active(true)
mark(false) → active(false)
active(f(z0)) → mark(if(z0, c, f(true)))
active(if(true, z0, z1)) → mark(z0)
active(if(false, z0, z1)) → mark(z1)
if(z0, mark(z1), z2) → if(z0, z1, z2)
if(mark(z0), 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)
f(active(z0)) → f(z0)
f(mark(z0)) → f(z0)

And the Tuples:

ACTIVE(if(true, z0, z1)) → c2(MARK(z0))
ACTIVE(if(false, z0, z1)) → c3(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))
ACTIVE(f(x0)) → c1(MARK(if(x0, c, f(true))), IF(x0, c, f(true)))
MARK(f(z0)) → c4(ACTIVE(f(z0)), F(mark(z0)), MARK(z0))
MARK(f(c)) → c4(ACTIVE(f(active(c))), F(mark(c)), MARK(c))
MARK(f(true)) → c4(ACTIVE(f(active(true))), F(mark(true)), MARK(true))
MARK(f(false)) → c4(ACTIVE(f(active(false))), F(mark(false)), MARK(false))
MARK(if(z0, x1, z2)) → c5(ACTIVE(if(z0, mark(x1), z2)), IF(mark(z0), mark(x1), z2), MARK(z0), MARK(x1))
MARK(if(x0, z1, z2)) → c5(ACTIVE(if(mark(x0), z1, z2)), IF(mark(x0), mark(z1), z2), MARK(x0), MARK(z1))
MARK(if(x0, f(z0), x2)) → c5(ACTIVE(if(mark(x0), active(f(mark(z0))), x2)), IF(mark(x0), mark(f(z0)), x2), MARK(x0), MARK(f(z0)))
MARK(if(x0, if(z0, z1, z2), x2)) → c5(ACTIVE(if(mark(x0), active(if(mark(z0), mark(z1), z2)), x2)), IF(mark(x0), mark(if(z0, z1, z2)), x2), MARK(x0), MARK(if(z0, z1, z2)))
MARK(if(x0, c, x2)) → c5(ACTIVE(if(mark(x0), active(c), x2)), IF(mark(x0), mark(c), x2), MARK(x0), MARK(c))
MARK(if(x0, true, x2)) → c5(ACTIVE(if(mark(x0), active(true), x2)), IF(mark(x0), mark(true), x2), MARK(x0), MARK(true))
MARK(if(x0, false, x2)) → c5(ACTIVE(if(mark(x0), active(false), x2)), IF(mark(x0), mark(false), x2), MARK(x0), MARK(false))
MARK(if(f(z0), x1, x2)) → c5(ACTIVE(if(active(f(mark(z0))), mark(x1), x2)), IF(mark(f(z0)), mark(x1), x2), MARK(f(z0)), MARK(x1))
MARK(if(if(z0, z1, z2), x1, x2)) → c5(ACTIVE(if(active(if(mark(z0), mark(z1), z2)), mark(x1), x2)), IF(mark(if(z0, z1, z2)), mark(x1), x2), MARK(if(z0, z1, z2)), MARK(x1))
MARK(if(c, x1, x2)) → c5(ACTIVE(if(active(c), mark(x1), x2)), IF(mark(c), mark(x1), x2), MARK(c), MARK(x1))
MARK(if(true, x1, x2)) → c5(ACTIVE(if(active(true), mark(x1), x2)), IF(mark(true), mark(x1), x2), MARK(true), MARK(x1))
MARK(if(false, x1, x2)) → c5(ACTIVE(if(active(false), mark(x1), x2)), IF(mark(false), mark(x1), x2), MARK(false), MARK(x1))
MARK(if(x0, x1, x2)) → c5(IF(mark(x0), mark(x1), x2))
MARK(f(f(x0))) → c4(ACTIVE(f(f(mark(x0)))), F(mark(f(x0))), MARK(f(x0)))
MARK(f(f(x0))) → c4(ACTIVE(f(mark(if(mark(x0), c, f(true))))), F(mark(f(x0))), MARK(f(x0)))
MARK(f(f(z0))) → c4(ACTIVE(f(active(f(z0)))), F(mark(f(z0))), MARK(f(z0)))
MARK(f(f(f(z0)))) → c4(ACTIVE(f(active(f(active(f(mark(z0))))))), F(mark(f(f(z0)))), MARK(f(f(z0))))
MARK(f(f(if(z0, z1, z2)))) → c4(ACTIVE(f(active(f(active(if(mark(z0), mark(z1), z2)))))), F(mark(f(if(z0, z1, z2)))), MARK(f(if(z0, z1, z2))))
MARK(f(f(c))) → c4(ACTIVE(f(active(f(active(c))))), F(mark(f(c))), MARK(f(c)))
MARK(f(f(true))) → c4(ACTIVE(f(active(f(active(true))))), F(mark(f(true))), MARK(f(true)))
MARK(f(f(false))) → c4(ACTIVE(f(active(f(active(false))))), F(mark(f(false))), MARK(f(false)))
MARK(f(f(x0))) → c4(F(mark(f(x0))))
MARK(f(if(x0, x1, x2))) → c4(ACTIVE(f(if(mark(x0), mark(x1), x2))), F(mark(if(x0, x1, x2))), MARK(if(x0, x1, x2)))
MARK(f(if(z0, x1, z2))) → c4(ACTIVE(f(active(if(z0, mark(x1), z2)))), F(mark(if(z0, x1, z2))), MARK(if(z0, x1, z2)))
MARK(f(if(x0, z1, z2))) → c4(ACTIVE(f(active(if(mark(x0), z1, z2)))), F(mark(if(x0, z1, z2))), MARK(if(x0, z1, z2)))
MARK(f(if(x0, f(z0), x2))) → c4(ACTIVE(f(active(if(mark(x0), active(f(mark(z0))), x2)))), F(mark(if(x0, f(z0), x2))), MARK(if(x0, f(z0), x2)))
MARK(f(if(x0, if(z0, z1, z2), x2))) → c4(ACTIVE(f(active(if(mark(x0), active(if(mark(z0), mark(z1), z2)), x2)))), F(mark(if(x0, if(z0, z1, z2), x2))), MARK(if(x0, if(z0, z1, z2), x2)))
MARK(f(if(x0, c, x2))) → c4(ACTIVE(f(active(if(mark(x0), active(c), x2)))), F(mark(if(x0, c, x2))), MARK(if(x0, c, x2)))
MARK(f(if(x0, true, x2))) → c4(ACTIVE(f(active(if(mark(x0), active(true), x2)))), F(mark(if(x0, true, x2))), MARK(if(x0, true, x2)))
MARK(f(if(x0, false, x2))) → c4(ACTIVE(f(active(if(mark(x0), active(false), x2)))), F(mark(if(x0, false, x2))), MARK(if(x0, false, x2)))
MARK(f(if(f(z0), x1, x2))) → c4(ACTIVE(f(active(if(active(f(mark(z0))), mark(x1), x2)))), F(mark(if(f(z0), x1, x2))), MARK(if(f(z0), x1, x2)))
MARK(f(if(if(z0, z1, z2), x1, x2))) → c4(ACTIVE(f(active(if(active(if(mark(z0), mark(z1), z2)), mark(x1), x2)))), F(mark(if(if(z0, z1, z2), x1, x2))), MARK(if(if(z0, z1, z2), x1, x2)))
MARK(f(if(c, x1, x2))) → c4(ACTIVE(f(active(if(active(c), mark(x1), x2)))), F(mark(if(c, x1, x2))), MARK(if(c, x1, x2)))
MARK(f(if(true, x1, x2))) → c4(ACTIVE(f(active(if(active(true), mark(x1), x2)))), F(mark(if(true, x1, x2))), MARK(if(true, x1, x2)))
MARK(f(if(false, x1, x2))) → c4(ACTIVE(f(active(if(active(false), mark(x1), x2)))), F(mark(if(false, x1, x2))), MARK(if(false, x1, x2)))
MARK(f(if(x0, x1, x2))) → c4(F(mark(if(x0, x1, x2))), MARK(if(x0, x1, x2)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVE(x₁)) = x₁
POL(F(x₁)) = 0
POL(IF(x₁, x₂, x₃)) = 0
POL(MARK(x₁)) = [2]x₁
POL(active(x₁)) = x₁
POL(c) = 0
POL(c1(x₁, x₂)) = x₁ + x₂
POL(c10(x₁)) = x₁
POL(c11(x₁)) = x₁
POL(c12(x₁)) = x₁
POL(c13(x₁)) = x₁
POL(c14(x₁)) = x₁
POL(c15(x₁)) = x₁
POL(c16(x₁)) = x₁
POL(c2(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(c4(x₁)) = x₁
POL(c4(x₁, x₂)) = x₁ + x₂
POL(c4(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c5(x₁)) = x₁
POL(c5(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(c9(x₁)) = x₁
POL(f(x₁)) = [4]x₁
POL(false) = [4]
POL(if(x₁, x₂, x₃)) = [2]x₁ + [2]x₂ + [4]x₃
POL(mark(x₁)) = x₁
POL(true) = 0

(28) 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(if(true, z0, z1)) → c2(MARK(z0))
ACTIVE(if(false, z0, z1)) → c3(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))
ACTIVE(f(x0)) → c1(MARK(if(x0, c, f(true))), IF(x0, c, f(true)))
MARK(f(z0)) → c4(ACTIVE(f(z0)), F(mark(z0)), MARK(z0))
MARK(f(c)) → c4(ACTIVE(f(active(c))), F(mark(c)), MARK(c))
MARK(f(true)) → c4(ACTIVE(f(active(true))), F(mark(true)), MARK(true))
MARK(f(false)) → c4(ACTIVE(f(active(false))), F(mark(false)), MARK(false))
MARK(if(z0, x1, z2)) → c5(ACTIVE(if(z0, mark(x1), z2)), IF(mark(z0), mark(x1), z2), MARK(z0), MARK(x1))
MARK(if(x0, z1, z2)) → c5(ACTIVE(if(mark(x0), z1, z2)), IF(mark(x0), mark(z1), z2), MARK(x0), MARK(z1))
MARK(if(x0, f(z0), x2)) → c5(ACTIVE(if(mark(x0), active(f(mark(z0))), x2)), IF(mark(x0), mark(f(z0)), x2), MARK(x0), MARK(f(z0)))
MARK(if(x0, if(z0, z1, z2), x2)) → c5(ACTIVE(if(mark(x0), active(if(mark(z0), mark(z1), z2)), x2)), IF(mark(x0), mark(if(z0, z1, z2)), x2), MARK(x0), MARK(if(z0, z1, z2)))
MARK(if(x0, c, x2)) → c5(ACTIVE(if(mark(x0), active(c), x2)), IF(mark(x0), mark(c), x2), MARK(x0), MARK(c))
MARK(if(x0, true, x2)) → c5(ACTIVE(if(mark(x0), active(true), x2)), IF(mark(x0), mark(true), x2), MARK(x0), MARK(true))
MARK(if(x0, false, x2)) → c5(ACTIVE(if(mark(x0), active(false), x2)), IF(mark(x0), mark(false), x2), MARK(x0), MARK(false))
MARK(if(f(z0), x1, x2)) → c5(ACTIVE(if(active(f(mark(z0))), mark(x1), x2)), IF(mark(f(z0)), mark(x1), x2), MARK(f(z0)), MARK(x1))
MARK(if(if(z0, z1, z2), x1, x2)) → c5(ACTIVE(if(active(if(mark(z0), mark(z1), z2)), mark(x1), x2)), IF(mark(if(z0, z1, z2)), mark(x1), x2), MARK(if(z0, z1, z2)), MARK(x1))
MARK(if(c, x1, x2)) → c5(ACTIVE(if(active(c), mark(x1), x2)), IF(mark(c), mark(x1), x2), MARK(c), MARK(x1))
MARK(if(true, x1, x2)) → c5(ACTIVE(if(active(true), mark(x1), x2)), IF(mark(true), mark(x1), x2), MARK(true), MARK(x1))
MARK(if(false, x1, x2)) → c5(ACTIVE(if(active(false), mark(x1), x2)), IF(mark(false), mark(x1), x2), MARK(false), MARK(x1))
MARK(if(x0, x1, x2)) → c5(IF(mark(x0), mark(x1), x2))
MARK(f(f(x0))) → c4(ACTIVE(f(f(mark(x0)))), F(mark(f(x0))), MARK(f(x0)))
MARK(f(f(x0))) → c4(ACTIVE(f(mark(if(mark(x0), c, f(true))))), F(mark(f(x0))), MARK(f(x0)))
MARK(f(f(z0))) → c4(ACTIVE(f(active(f(z0)))), F(mark(f(z0))), MARK(f(z0)))
MARK(f(f(f(z0)))) → c4(ACTIVE(f(active(f(active(f(mark(z0))))))), F(mark(f(f(z0)))), MARK(f(f(z0))))
MARK(f(f(if(z0, z1, z2)))) → c4(ACTIVE(f(active(f(active(if(mark(z0), mark(z1), z2)))))), F(mark(f(if(z0, z1, z2)))), MARK(f(if(z0, z1, z2))))
MARK(f(f(c))) → c4(ACTIVE(f(active(f(active(c))))), F(mark(f(c))), MARK(f(c)))
MARK(f(f(true))) → c4(ACTIVE(f(active(f(active(true))))), F(mark(f(true))), MARK(f(true)))
MARK(f(f(false))) → c4(ACTIVE(f(active(f(active(false))))), F(mark(f(false))), MARK(f(false)))
MARK(f(f(x0))) → c4(F(mark(f(x0))))
MARK(f(if(x0, x1, x2))) → c4(ACTIVE(f(if(mark(x0), mark(x1), x2))), F(mark(if(x0, x1, x2))), MARK(if(x0, x1, x2)))
MARK(f(if(z0, x1, z2))) → c4(ACTIVE(f(active(if(z0, mark(x1), z2)))), F(mark(if(z0, x1, z2))), MARK(if(z0, x1, z2)))
MARK(f(if(x0, z1, z2))) → c4(ACTIVE(f(active(if(mark(x0), z1, z2)))), F(mark(if(x0, z1, z2))), MARK(if(x0, z1, z2)))
MARK(f(if(x0, f(z0), x2))) → c4(ACTIVE(f(active(if(mark(x0), active(f(mark(z0))), x2)))), F(mark(if(x0, f(z0), x2))), MARK(if(x0, f(z0), x2)))
MARK(f(if(x0, if(z0, z1, z2), x2))) → c4(ACTIVE(f(active(if(mark(x0), active(if(mark(z0), mark(z1), z2)), x2)))), F(mark(if(x0, if(z0, z1, z2), x2))), MARK(if(x0, if(z0, z1, z2), x2)))
MARK(f(if(x0, c, x2))) → c4(ACTIVE(f(active(if(mark(x0), active(c), x2)))), F(mark(if(x0, c, x2))), MARK(if(x0, c, x2)))
MARK(f(if(x0, true, x2))) → c4(ACTIVE(f(active(if(mark(x0), active(true), x2)))), F(mark(if(x0, true, x2))), MARK(if(x0, true, x2)))
MARK(f(if(x0, false, x2))) → c4(ACTIVE(f(active(if(mark(x0), active(false), x2)))), F(mark(if(x0, false, x2))), MARK(if(x0, false, x2)))
MARK(f(if(f(z0), x1, x2))) → c4(ACTIVE(f(active(if(active(f(mark(z0))), mark(x1), x2)))), F(mark(if(f(z0), x1, x2))), MARK(if(f(z0), x1, x2)))
MARK(f(if(if(z0, z1, z2), x1, x2))) → c4(ACTIVE(f(active(if(active(if(mark(z0), mark(z1), z2)), mark(x1), x2)))), F(mark(if(if(z0, z1, z2), x1, x2))), MARK(if(if(z0, z1, z2), x1, x2)))
MARK(f(if(c, x1, x2))) → c4(ACTIVE(f(active(if(active(c), mark(x1), x2)))), F(mark(if(c, x1, x2))), MARK(if(c, x1, x2)))
MARK(f(if(true, x1, x2))) → c4(ACTIVE(f(active(if(active(true), mark(x1), x2)))), F(mark(if(true, x1, x2))), MARK(if(true, x1, x2)))
MARK(f(if(false, x1, x2))) → c4(ACTIVE(f(active(if(active(false), mark(x1), x2)))), F(mark(if(false, x1, x2))), MARK(if(false, x1, x2)))
MARK(f(if(x0, x1, x2))) → c4(F(mark(if(x0, x1, x2))), MARK(if(x0, x1, x2)))

S tuples:

ACTIVE(if(true, z0, z1)) → c2(MARK(z0))
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))
ACTIVE(f(x0)) → c1(MARK(if(x0, c, f(true))), IF(x0, c, f(true)))
MARK(f(z0)) → c4(ACTIVE(f(z0)), F(mark(z0)), MARK(z0))
MARK(f(c)) → c4(ACTIVE(f(active(c))), F(mark(c)), MARK(c))
MARK(f(true)) → c4(ACTIVE(f(active(true))), F(mark(true)), MARK(true))
MARK(if(z0, x1, z2)) → c5(ACTIVE(if(z0, mark(x1), z2)), IF(mark(z0), mark(x1), z2), MARK(z0), MARK(x1))
MARK(if(x0, z1, z2)) → c5(ACTIVE(if(mark(x0), z1, z2)), IF(mark(x0), mark(z1), z2), MARK(x0), MARK(z1))
MARK(if(x0, f(z0), x2)) → c5(ACTIVE(if(mark(x0), active(f(mark(z0))), x2)), IF(mark(x0), mark(f(z0)), x2), MARK(x0), MARK(f(z0)))
MARK(if(x0, if(z0, z1, z2), x2)) → c5(ACTIVE(if(mark(x0), active(if(mark(z0), mark(z1), z2)), x2)), IF(mark(x0), mark(if(z0, z1, z2)), x2), MARK(x0), MARK(if(z0, z1, z2)))
MARK(if(x0, c, x2)) → c5(ACTIVE(if(mark(x0), active(c), x2)), IF(mark(x0), mark(c), x2), MARK(x0), MARK(c))
MARK(if(x0, true, x2)) → c5(ACTIVE(if(mark(x0), active(true), x2)), IF(mark(x0), mark(true), x2), MARK(x0), MARK(true))
MARK(if(f(z0), x1, x2)) → c5(ACTIVE(if(active(f(mark(z0))), mark(x1), x2)), IF(mark(f(z0)), mark(x1), x2), MARK(f(z0)), MARK(x1))
MARK(if(if(z0, z1, z2), x1, x2)) → c5(ACTIVE(if(active(if(mark(z0), mark(z1), z2)), mark(x1), x2)), IF(mark(if(z0, z1, z2)), mark(x1), x2), MARK(if(z0, z1, z2)), MARK(x1))
MARK(if(c, x1, x2)) → c5(ACTIVE(if(active(c), mark(x1), x2)), IF(mark(c), mark(x1), x2), MARK(c), MARK(x1))
MARK(if(true, x1, x2)) → c5(ACTIVE(if(active(true), mark(x1), x2)), IF(mark(true), mark(x1), x2), MARK(true), MARK(x1))
MARK(if(x0, x1, x2)) → c5(IF(mark(x0), mark(x1), x2))
MARK(f(f(x0))) → c4(ACTIVE(f(f(mark(x0)))), F(mark(f(x0))), MARK(f(x0)))
MARK(f(f(x0))) → c4(ACTIVE(f(mark(if(mark(x0), c, f(true))))), F(mark(f(x0))), MARK(f(x0)))
MARK(f(f(z0))) → c4(ACTIVE(f(active(f(z0)))), F(mark(f(z0))), MARK(f(z0)))
MARK(f(f(f(z0)))) → c4(ACTIVE(f(active(f(active(f(mark(z0))))))), F(mark(f(f(z0)))), MARK(f(f(z0))))
MARK(f(f(if(z0, z1, z2)))) → c4(ACTIVE(f(active(f(active(if(mark(z0), mark(z1), z2)))))), F(mark(f(if(z0, z1, z2)))), MARK(f(if(z0, z1, z2))))
MARK(f(f(c))) → c4(ACTIVE(f(active(f(active(c))))), F(mark(f(c))), MARK(f(c)))
MARK(f(f(true))) → c4(ACTIVE(f(active(f(active(true))))), F(mark(f(true))), MARK(f(true)))
MARK(f(f(x0))) → c4(F(mark(f(x0))))
MARK(f(if(x0, x1, x2))) → c4(ACTIVE(f(if(mark(x0), mark(x1), x2))), F(mark(if(x0, x1, x2))), MARK(if(x0, x1, x2)))
MARK(f(if(z0, x1, z2))) → c4(ACTIVE(f(active(if(z0, mark(x1), z2)))), F(mark(if(z0, x1, z2))), MARK(if(z0, x1, z2)))
MARK(f(if(x0, z1, z2))) → c4(ACTIVE(f(active(if(mark(x0), z1, z2)))), F(mark(if(x0, z1, z2))), MARK(if(x0, z1, z2)))
MARK(f(if(x0, f(z0), x2))) → c4(ACTIVE(f(active(if(mark(x0), active(f(mark(z0))), x2)))), F(mark(if(x0, f(z0), x2))), MARK(if(x0, f(z0), x2)))
MARK(f(if(x0, if(z0, z1, z2), x2))) → c4(ACTIVE(f(active(if(mark(x0), active(if(mark(z0), mark(z1), z2)), x2)))), F(mark(if(x0, if(z0, z1, z2), x2))), MARK(if(x0, if(z0, z1, z2), x2)))
MARK(f(if(x0, c, x2))) → c4(ACTIVE(f(active(if(mark(x0), active(c), x2)))), F(mark(if(x0, c, x2))), MARK(if(x0, c, x2)))
MARK(f(if(x0, true, x2))) → c4(ACTIVE(f(active(if(mark(x0), active(true), x2)))), F(mark(if(x0, true, x2))), MARK(if(x0, true, x2)))
MARK(f(if(f(z0), x1, x2))) → c4(ACTIVE(f(active(if(active(f(mark(z0))), mark(x1), x2)))), F(mark(if(f(z0), x1, x2))), MARK(if(f(z0), x1, x2)))
MARK(f(if(if(z0, z1, z2), x1, x2))) → c4(ACTIVE(f(active(if(active(if(mark(z0), mark(z1), z2)), mark(x1), x2)))), F(mark(if(if(z0, z1, z2), x1, x2))), MARK(if(if(z0, z1, z2), x1, x2)))
MARK(f(if(c, x1, x2))) → c4(ACTIVE(f(active(if(active(c), mark(x1), x2)))), F(mark(if(c, x1, x2))), MARK(if(c, x1, x2)))
MARK(f(if(true, x1, x2))) → c4(ACTIVE(f(active(if(active(true), mark(x1), x2)))), F(mark(if(true, x1, x2))), MARK(if(true, x1, x2)))
MARK(f(if(x0, x1, x2))) → c4(F(mark(if(x0, x1, x2))), MARK(if(x0, x1, x2)))

K tuples:

ACTIVE(if(false, z0, z1)) → c3(MARK(z1))
MARK(f(false)) → c4(ACTIVE(f(active(false))), F(mark(false)), MARK(false))
MARK(if(x0, false, x2)) → c5(ACTIVE(if(mark(x0), active(false), x2)), IF(mark(x0), mark(false), x2), MARK(x0), MARK(false))
MARK(if(false, x1, x2)) → c5(ACTIVE(if(active(false), mark(x1), x2)), IF(mark(false), mark(x1), x2), MARK(false), MARK(x1))
MARK(f(f(false))) → c4(ACTIVE(f(active(f(active(false))))), F(mark(f(false))), MARK(f(false)))
MARK(f(if(x0, false, x2))) → c4(ACTIVE(f(active(if(mark(x0), active(false), x2)))), F(mark(if(x0, false, x2))), MARK(if(x0, false, x2)))
MARK(f(if(false, x1, x2))) → c4(ACTIVE(f(active(if(active(false), mark(x1), x2)))), F(mark(if(false, x1, x2))), MARK(if(false, x1, x2)))

Defined Rule Symbols:

active, mark, f, if

Defined Pair Symbols:

ACTIVE, F, IF, MARK

Compound Symbols:

c2, c3, c9, c10, c11, c12, c13, c14, c15, c16, c1, c4, c5, c5, c4, c4

(29) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace MARK(f(c)) → c4(ACTIVE(f(active(c))), F(mark(c)), MARK(c)) by

MARK(f(c)) → c4(ACTIVE(f(c)), F(mark(c)), MARK(c))
MARK(f(c)) → c4(F(mark(c)))

(30) 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(if(true, z0, z1)) → c2(MARK(z0))
ACTIVE(if(false, z0, z1)) → c3(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))
ACTIVE(f(x0)) → c1(MARK(if(x0, c, f(true))), IF(x0, c, f(true)))
MARK(f(z0)) → c4(ACTIVE(f(z0)), F(mark(z0)), MARK(z0))
MARK(f(true)) → c4(ACTIVE(f(active(true))), F(mark(true)), MARK(true))
MARK(f(false)) → c4(ACTIVE(f(active(false))), F(mark(false)), MARK(false))
MARK(if(z0, x1, z2)) → c5(ACTIVE(if(z0, mark(x1), z2)), IF(mark(z0), mark(x1), z2), MARK(z0), MARK(x1))
MARK(if(x0, z1, z2)) → c5(ACTIVE(if(mark(x0), z1, z2)), IF(mark(x0), mark(z1), z2), MARK(x0), MARK(z1))
MARK(if(x0, f(z0), x2)) → c5(ACTIVE(if(mark(x0), active(f(mark(z0))), x2)), IF(mark(x0), mark(f(z0)), x2), MARK(x0), MARK(f(z0)))
MARK(if(x0, if(z0, z1, z2), x2)) → c5(ACTIVE(if(mark(x0), active(if(mark(z0), mark(z1), z2)), x2)), IF(mark(x0), mark(if(z0, z1, z2)), x2), MARK(x0), MARK(if(z0, z1, z2)))
MARK(if(x0, c, x2)) → c5(ACTIVE(if(mark(x0), active(c), x2)), IF(mark(x0), mark(c), x2), MARK(x0), MARK(c))
MARK(if(x0, true, x2)) → c5(ACTIVE(if(mark(x0), active(true), x2)), IF(mark(x0), mark(true), x2), MARK(x0), MARK(true))
MARK(if(x0, false, x2)) → c5(ACTIVE(if(mark(x0), active(false), x2)), IF(mark(x0), mark(false), x2), MARK(x0), MARK(false))
MARK(if(f(z0), x1, x2)) → c5(ACTIVE(if(active(f(mark(z0))), mark(x1), x2)), IF(mark(f(z0)), mark(x1), x2), MARK(f(z0)), MARK(x1))
MARK(if(if(z0, z1, z2), x1, x2)) → c5(ACTIVE(if(active(if(mark(z0), mark(z1), z2)), mark(x1), x2)), IF(mark(if(z0, z1, z2)), mark(x1), x2), MARK(if(z0, z1, z2)), MARK(x1))
MARK(if(c, x1, x2)) → c5(ACTIVE(if(active(c), mark(x1), x2)), IF(mark(c), mark(x1), x2), MARK(c), MARK(x1))
MARK(if(true, x1, x2)) → c5(ACTIVE(if(active(true), mark(x1), x2)), IF(mark(true), mark(x1), x2), MARK(true), MARK(x1))
MARK(if(false, x1, x2)) → c5(ACTIVE(if(active(false), mark(x1), x2)), IF(mark(false), mark(x1), x2), MARK(false), MARK(x1))
MARK(if(x0, x1, x2)) → c5(IF(mark(x0), mark(x1), x2))
MARK(f(f(x0))) → c4(ACTIVE(f(f(mark(x0)))), F(mark(f(x0))), MARK(f(x0)))
MARK(f(f(x0))) → c4(ACTIVE(f(mark(if(mark(x0), c, f(true))))), F(mark(f(x0))), MARK(f(x0)))
MARK(f(f(z0))) → c4(ACTIVE(f(active(f(z0)))), F(mark(f(z0))), MARK(f(z0)))
MARK(f(f(f(z0)))) → c4(ACTIVE(f(active(f(active(f(mark(z0))))))), F(mark(f(f(z0)))), MARK(f(f(z0))))
MARK(f(f(if(z0, z1, z2)))) → c4(ACTIVE(f(active(f(active(if(mark(z0), mark(z1), z2)))))), F(mark(f(if(z0, z1, z2)))), MARK(f(if(z0, z1, z2))))
MARK(f(f(c))) → c4(ACTIVE(f(active(f(active(c))))), F(mark(f(c))), MARK(f(c)))
MARK(f(f(true))) → c4(ACTIVE(f(active(f(active(true))))), F(mark(f(true))), MARK(f(true)))
MARK(f(f(false))) → c4(ACTIVE(f(active(f(active(false))))), F(mark(f(false))), MARK(f(false)))
MARK(f(f(x0))) → c4(F(mark(f(x0))))
MARK(f(if(x0, x1, x2))) → c4(ACTIVE(f(if(mark(x0), mark(x1), x2))), F(mark(if(x0, x1, x2))), MARK(if(x0, x1, x2)))
MARK(f(if(z0, x1, z2))) → c4(ACTIVE(f(active(if(z0, mark(x1), z2)))), F(mark(if(z0, x1, z2))), MARK(if(z0, x1, z2)))
MARK(f(if(x0, z1, z2))) → c4(ACTIVE(f(active(if(mark(x0), z1, z2)))), F(mark(if(x0, z1, z2))), MARK(if(x0, z1, z2)))
MARK(f(if(x0, f(z0), x2))) → c4(ACTIVE(f(active(if(mark(x0), active(f(mark(z0))), x2)))), F(mark(if(x0, f(z0), x2))), MARK(if(x0, f(z0), x2)))
MARK(f(if(x0, if(z0, z1, z2), x2))) → c4(ACTIVE(f(active(if(mark(x0), active(if(mark(z0), mark(z1), z2)), x2)))), F(mark(if(x0, if(z0, z1, z2), x2))), MARK(if(x0, if(z0, z1, z2), x2)))
MARK(f(if(x0, c, x2))) → c4(ACTIVE(f(active(if(mark(x0), active(c), x2)))), F(mark(if(x0, c, x2))), MARK(if(x0, c, x2)))
MARK(f(if(x0, true, x2))) → c4(ACTIVE(f(active(if(mark(x0), active(true), x2)))), F(mark(if(x0, true, x2))), MARK(if(x0, true, x2)))
MARK(f(if(x0, false, x2))) → c4(ACTIVE(f(active(if(mark(x0), active(false), x2)))), F(mark(if(x0, false, x2))), MARK(if(x0, false, x2)))
MARK(f(if(f(z0), x1, x2))) → c4(ACTIVE(f(active(if(active(f(mark(z0))), mark(x1), x2)))), F(mark(if(f(z0), x1, x2))), MARK(if(f(z0), x1, x2)))
MARK(f(if(if(z0, z1, z2), x1, x2))) → c4(ACTIVE(f(active(if(active(if(mark(z0), mark(z1), z2)), mark(x1), x2)))), F(mark(if(if(z0, z1, z2), x1, x2))), MARK(if(if(z0, z1, z2), x1, x2)))
MARK(f(if(c, x1, x2))) → c4(ACTIVE(f(active(if(active(c), mark(x1), x2)))), F(mark(if(c, x1, x2))), MARK(if(c, x1, x2)))
MARK(f(if(true, x1, x2))) → c4(ACTIVE(f(active(if(active(true), mark(x1), x2)))), F(mark(if(true, x1, x2))), MARK(if(true, x1, x2)))
MARK(f(if(false, x1, x2))) → c4(ACTIVE(f(active(if(active(false), mark(x1), x2)))), F(mark(if(false, x1, x2))), MARK(if(false, x1, x2)))
MARK(f(if(x0, x1, x2))) → c4(F(mark(if(x0, x1, x2))), MARK(if(x0, x1, x2)))
MARK(f(c)) → c4(ACTIVE(f(c)), F(mark(c)), MARK(c))
MARK(f(c)) → c4(F(mark(c)))

S tuples:

ACTIVE(if(true, z0, z1)) → c2(MARK(z0))
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))
ACTIVE(f(x0)) → c1(MARK(if(x0, c, f(true))), IF(x0, c, f(true)))
MARK(f(z0)) → c4(ACTIVE(f(z0)), F(mark(z0)), MARK(z0))
MARK(f(true)) → c4(ACTIVE(f(active(true))), F(mark(true)), MARK(true))
MARK(if(z0, x1, z2)) → c5(ACTIVE(if(z0, mark(x1), z2)), IF(mark(z0), mark(x1), z2), MARK(z0), MARK(x1))
MARK(if(x0, z1, z2)) → c5(ACTIVE(if(mark(x0), z1, z2)), IF(mark(x0), mark(z1), z2), MARK(x0), MARK(z1))
MARK(if(x0, f(z0), x2)) → c5(ACTIVE(if(mark(x0), active(f(mark(z0))), x2)), IF(mark(x0), mark(f(z0)), x2), MARK(x0), MARK(f(z0)))
MARK(if(x0, if(z0, z1, z2), x2)) → c5(ACTIVE(if(mark(x0), active(if(mark(z0), mark(z1), z2)), x2)), IF(mark(x0), mark(if(z0, z1, z2)), x2), MARK(x0), MARK(if(z0, z1, z2)))
MARK(if(x0, c, x2)) → c5(ACTIVE(if(mark(x0), active(c), x2)), IF(mark(x0), mark(c), x2), MARK(x0), MARK(c))
MARK(if(x0, true, x2)) → c5(ACTIVE(if(mark(x0), active(true), x2)), IF(mark(x0), mark(true), x2), MARK(x0), MARK(true))
MARK(if(f(z0), x1, x2)) → c5(ACTIVE(if(active(f(mark(z0))), mark(x1), x2)), IF(mark(f(z0)), mark(x1), x2), MARK(f(z0)), MARK(x1))
MARK(if(if(z0, z1, z2), x1, x2)) → c5(ACTIVE(if(active(if(mark(z0), mark(z1), z2)), mark(x1), x2)), IF(mark(if(z0, z1, z2)), mark(x1), x2), MARK(if(z0, z1, z2)), MARK(x1))
MARK(if(c, x1, x2)) → c5(ACTIVE(if(active(c), mark(x1), x2)), IF(mark(c), mark(x1), x2), MARK(c), MARK(x1))
MARK(if(true, x1, x2)) → c5(ACTIVE(if(active(true), mark(x1), x2)), IF(mark(true), mark(x1), x2), MARK(true), MARK(x1))
MARK(if(x0, x1, x2)) → c5(IF(mark(x0), mark(x1), x2))
MARK(f(f(x0))) → c4(ACTIVE(f(f(mark(x0)))), F(mark(f(x0))), MARK(f(x0)))
MARK(f(f(x0))) → c4(ACTIVE(f(mark(if(mark(x0), c, f(true))))), F(mark(f(x0))), MARK(f(x0)))
MARK(f(f(z0))) → c4(ACTIVE(f(active(f(z0)))), F(mark(f(z0))), MARK(f(z0)))
MARK(f(f(f(z0)))) → c4(ACTIVE(f(active(f(active(f(mark(z0))))))), F(mark(f(f(z0)))), MARK(f(f(z0))))
MARK(f(f(if(z0, z1, z2)))) → c4(ACTIVE(f(active(f(active(if(mark(z0), mark(z1), z2)))))), F(mark(f(if(z0, z1, z2)))), MARK(f(if(z0, z1, z2))))
MARK(f(f(c))) → c4(ACTIVE(f(active(f(active(c))))), F(mark(f(c))), MARK(f(c)))
MARK(f(f(true))) → c4(ACTIVE(f(active(f(active(true))))), F(mark(f(true))), MARK(f(true)))
MARK(f(f(x0))) → c4(F(mark(f(x0))))
MARK(f(if(x0, x1, x2))) → c4(ACTIVE(f(if(mark(x0), mark(x1), x2))), F(mark(if(x0, x1, x2))), MARK(if(x0, x1, x2)))
MARK(f(if(z0, x1, z2))) → c4(ACTIVE(f(active(if(z0, mark(x1), z2)))), F(mark(if(z0, x1, z2))), MARK(if(z0, x1, z2)))
MARK(f(if(x0, z1, z2))) → c4(ACTIVE(f(active(if(mark(x0), z1, z2)))), F(mark(if(x0, z1, z2))), MARK(if(x0, z1, z2)))
MARK(f(if(x0, f(z0), x2))) → c4(ACTIVE(f(active(if(mark(x0), active(f(mark(z0))), x2)))), F(mark(if(x0, f(z0), x2))), MARK(if(x0, f(z0), x2)))
MARK(f(if(x0, if(z0, z1, z2), x2))) → c4(ACTIVE(f(active(if(mark(x0), active(if(mark(z0), mark(z1), z2)), x2)))), F(mark(if(x0, if(z0, z1, z2), x2))), MARK(if(x0, if(z0, z1, z2), x2)))
MARK(f(if(x0, c, x2))) → c4(ACTIVE(f(active(if(mark(x0), active(c), x2)))), F(mark(if(x0, c, x2))), MARK(if(x0, c, x2)))
MARK(f(if(x0, true, x2))) → c4(ACTIVE(f(active(if(mark(x0), active(true), x2)))), F(mark(if(x0, true, x2))), MARK(if(x0, true, x2)))
MARK(f(if(f(z0), x1, x2))) → c4(ACTIVE(f(active(if(active(f(mark(z0))), mark(x1), x2)))), F(mark(if(f(z0), x1, x2))), MARK(if(f(z0), x1, x2)))
MARK(f(if(if(z0, z1, z2), x1, x2))) → c4(ACTIVE(f(active(if(active(if(mark(z0), mark(z1), z2)), mark(x1), x2)))), F(mark(if(if(z0, z1, z2), x1, x2))), MARK(if(if(z0, z1, z2), x1, x2)))
MARK(f(if(c, x1, x2))) → c4(ACTIVE(f(active(if(active(c), mark(x1), x2)))), F(mark(if(c, x1, x2))), MARK(if(c, x1, x2)))
MARK(f(if(true, x1, x2))) → c4(ACTIVE(f(active(if(active(true), mark(x1), x2)))), F(mark(if(true, x1, x2))), MARK(if(true, x1, x2)))
MARK(f(if(x0, x1, x2))) → c4(F(mark(if(x0, x1, x2))), MARK(if(x0, x1, x2)))
MARK(f(c)) → c4(ACTIVE(f(c)), F(mark(c)), MARK(c))
MARK(f(c)) → c4(F(mark(c)))

K tuples:

ACTIVE(if(false, z0, z1)) → c3(MARK(z1))
MARK(f(false)) → c4(ACTIVE(f(active(false))), F(mark(false)), MARK(false))
MARK(if(x0, false, x2)) → c5(ACTIVE(if(mark(x0), active(false), x2)), IF(mark(x0), mark(false), x2), MARK(x0), MARK(false))
MARK(if(false, x1, x2)) → c5(ACTIVE(if(active(false), mark(x1), x2)), IF(mark(false), mark(x1), x2), MARK(false), MARK(x1))
MARK(f(f(false))) → c4(ACTIVE(f(active(f(active(false))))), F(mark(f(false))), MARK(f(false)))
MARK(f(if(x0, false, x2))) → c4(ACTIVE(f(active(if(mark(x0), active(false), x2)))), F(mark(if(x0, false, x2))), MARK(if(x0, false, x2)))
MARK(f(if(false, x1, x2))) → c4(ACTIVE(f(active(if(active(false), mark(x1), x2)))), F(mark(if(false, x1, x2))), MARK(if(false, x1, x2)))

Defined Rule Symbols:

active, mark, f, if

Defined Pair Symbols:

ACTIVE, F, IF, MARK

Compound Symbols:

c2, c3, c9, c10, c11, c12, c13, c14, c15, c16, c1, c4, c5, c5, c4, c4

(31) CpxTrsMatchBoundsTAProof (EQUIVALENT transformation)

A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 1.

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by:
final states : [1, 2, 3, 4]
transitions:
c0() → 0
true0() → 0
false0() → 0
active0(0) → 1
mark0(0) → 2
f0(0) → 3
if0(0, 0, 0) → 4
c1() → 5
active1(5) → 2
true1() → 6
active1(6) → 2
false1() → 7
active1(7) → 2

(0) Obligation:

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

(2) Obligation:

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

(4) Obligation:

(5) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

(6) Obligation:

(7) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

(8) Obligation:

(9) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

(10) Obligation:

(11) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

(12) Obligation:

(13) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

(14) Obligation:

(15) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

(16) Obligation:

(17) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

(18) Obligation:

(19) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

(20) Obligation:

(21) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

(22) Obligation:

(23) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

(24) Obligation:

(25) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

(26) Obligation:

(27) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

(28) Obligation:

(29) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

(30) Obligation:

(31) CpxTrsMatchBoundsTAProof (EQUIVALENT transformation)

(32) BOUNDS(O(1), O(n^1))