The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^2)).

(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)
active(f(X)) → f(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(if(X1, X2, X3)) → if(X1, active(X2), X3)
f(mark(X)) → mark(f(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
if(X1, mark(X2), X3) → mark(if(X1, X2, X3))
proper(f(X)) → f(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(c) → ok(c)
proper(true) → ok(true)
proper(false) → ok(false)
f(ok(X)) → ok(f(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

ACTIVE(f(z0)) → c1(IF(z0, c, f(true)), F(true))
ACTIVE(f(z0)) → c4(F(active(z0)), ACTIVE(z0))
ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1))
F(mark(z0)) → c7(F(z0))
F(ok(z0)) → c8(F(z0))
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2))
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))
PROPER(f(z0)) → c12(F(proper(z0)), PROPER(z0))
PROPER(if(z0, z1, z2)) → c13(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
TOP(mark(z0)) → c17(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c18(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(f(z0)) → c1(IF(z0, c, f(true)), F(true))
ACTIVE(f(z0)) → c4(F(active(z0)), ACTIVE(z0))
ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1))
F(mark(z0)) → c7(F(z0))
F(ok(z0)) → c8(F(z0))
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2))
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))
PROPER(f(z0)) → c12(F(proper(z0)), PROPER(z0))
PROPER(if(z0, z1, z2)) → c13(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
TOP(mark(z0)) → c17(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c18(TOP(active(z0)), ACTIVE(z0))

K tuples:none
Defined Rule Symbols:

active, f, if, proper, top

Defined Pair Symbols:

ACTIVE, F, IF, PROPER, TOP

Compound Symbols:

c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c17, c18

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

Removed 1 trailing tuple parts

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

Tuples:

ACTIVE(f(z0)) → c4(F(active(z0)), ACTIVE(z0))
ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1))
F(mark(z0)) → c7(F(z0))
F(ok(z0)) → c8(F(z0))
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2))
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))
PROPER(f(z0)) → c12(F(proper(z0)), PROPER(z0))
PROPER(if(z0, z1, z2)) → c13(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
TOP(mark(z0)) → c17(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c18(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c1(IF(z0, c, f(true)))

S tuples:

ACTIVE(f(z0)) → c4(F(active(z0)), ACTIVE(z0))
ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1))
F(mark(z0)) → c7(F(z0))
F(ok(z0)) → c8(F(z0))
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2))
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))
PROPER(f(z0)) → c12(F(proper(z0)), PROPER(z0))
PROPER(if(z0, z1, z2)) → c13(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
TOP(mark(z0)) → c17(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c18(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c1(IF(z0, c, f(true)))

K tuples:none
Defined Rule Symbols:

active, f, if, proper, top

Defined Pair Symbols:

ACTIVE, F, IF, PROPER, TOP

Compound Symbols:

c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c17, c18, c1

(5) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

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

TOP(mark(z0)) → c17(TOP(proper(z0)), PROPER(z0))

We considered the (Usable) Rules:

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

And the Tuples:

ACTIVE(f(z0)) → c4(F(active(z0)), ACTIVE(z0))
ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1))
F(mark(z0)) → c7(F(z0))
F(ok(z0)) → c8(F(z0))
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2))
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))
PROPER(f(z0)) → c12(F(proper(z0)), PROPER(z0))
PROPER(if(z0, z1, z2)) → c13(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
TOP(mark(z0)) → c17(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c18(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c1(IF(z0, c, f(true)))

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

POL(ACTIVE(x₁)) = 0   
POL(F(x₁)) = 0   
POL(IF(x₁, x₂, x₃)) = 0   
POL(PROPER(x₁)) = 0   
POL(TOP(x₁)) = x₁   
POL(active(x₁)) = x₁   
POL(c) = 0   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁, x₂)) = x₁ + x₂   
POL(c13(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄   
POL(c17(x₁, x₂)) = x₁ + x₂   
POL(c18(x₁, x₂)) = x₁ + x₂   
POL(c4(x₁, x₂)) = x₁ + x₂   
POL(c5(x₁, x₂)) = x₁ + x₂   
POL(c6(x₁, x₂)) = x₁ + x₂   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(f(x₁)) = [2] + [2]x₁ + [2]x₁²   
POL(false) = [1]   
POL(if(x₁, x₂, x₃)) = [1] + x₂ + x₁·x₃ + [2]x₁²   
POL(mark(x₁)) = [1] + x₁   
POL(ok(x₁)) = x₁   
POL(proper(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)
active(f(z0)) → f(active(z0))
active(if(z0, z1, z2)) → if(active(z0), z1, z2)
active(if(z0, z1, z2)) → if(z0, active(z1), z2)
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
if(mark(z0), z1, z2) → mark(if(z0, z1, z2))
if(z0, mark(z1), z2) → mark(if(z0, z1, z2))
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2))
proper(f(z0)) → f(proper(z0))
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2))
proper(c) → ok(c)
proper(true) → ok(true)
proper(false) → ok(false)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(f(z0)) → c4(F(active(z0)), ACTIVE(z0))
ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1))
F(mark(z0)) → c7(F(z0))
F(ok(z0)) → c8(F(z0))
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2))
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))
PROPER(f(z0)) → c12(F(proper(z0)), PROPER(z0))
PROPER(if(z0, z1, z2)) → c13(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
TOP(mark(z0)) → c17(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c18(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c1(IF(z0, c, f(true)))

S tuples:

ACTIVE(f(z0)) → c4(F(active(z0)), ACTIVE(z0))
ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1))
F(mark(z0)) → c7(F(z0))
F(ok(z0)) → c8(F(z0))
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2))
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))
PROPER(f(z0)) → c12(F(proper(z0)), PROPER(z0))
PROPER(if(z0, z1, z2)) → c13(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
TOP(ok(z0)) → c18(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c1(IF(z0, c, f(true)))

K tuples:

TOP(mark(z0)) → c17(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, f, if, proper, top

Defined Pair Symbols:

ACTIVE, F, IF, PROPER, TOP

Compound Symbols:

c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c17, c18, c1

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

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

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

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

Tuples:

ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1))
F(mark(z0)) → c7(F(z0))
F(ok(z0)) → c8(F(z0))
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2))
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))
PROPER(f(z0)) → c12(F(proper(z0)), PROPER(z0))
PROPER(if(z0, z1, z2)) → c13(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
TOP(mark(z0)) → c17(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c18(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c1(IF(z0, c, f(true)))
ACTIVE(f(f(z0))) → c4(F(mark(if(z0, c, f(true)))), ACTIVE(f(z0)))
ACTIVE(f(if(true, z0, z1))) → c4(F(mark(z0)), ACTIVE(if(true, z0, z1)))
ACTIVE(f(if(false, z0, z1))) → c4(F(mark(z1)), ACTIVE(if(false, z0, z1)))
ACTIVE(f(f(z0))) → c4(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(z0, active(z1), z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(f(x0)) → c4

S tuples:

ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1))
F(mark(z0)) → c7(F(z0))
F(ok(z0)) → c8(F(z0))
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2))
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))
PROPER(f(z0)) → c12(F(proper(z0)), PROPER(z0))
PROPER(if(z0, z1, z2)) → c13(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
TOP(ok(z0)) → c18(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c1(IF(z0, c, f(true)))
ACTIVE(f(f(z0))) → c4(F(mark(if(z0, c, f(true)))), ACTIVE(f(z0)))
ACTIVE(f(if(true, z0, z1))) → c4(F(mark(z0)), ACTIVE(if(true, z0, z1)))
ACTIVE(f(if(false, z0, z1))) → c4(F(mark(z1)), ACTIVE(if(false, z0, z1)))
ACTIVE(f(f(z0))) → c4(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(z0, active(z1), z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(f(x0)) → c4

K tuples:

TOP(mark(z0)) → c17(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, f, if, proper, top

Defined Pair Symbols:

ACTIVE, F, IF, PROPER, TOP

Compound Symbols:

c5, c6, c7, c8, c9, c10, c11, c12, c13, c17, c18, c1, c4, c4

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

Removed 1 trailing nodes:

ACTIVE(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)
active(f(z0)) → f(active(z0))
active(if(z0, z1, z2)) → if(active(z0), z1, z2)
active(if(z0, z1, z2)) → if(z0, active(z1), z2)
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
if(mark(z0), z1, z2) → mark(if(z0, z1, z2))
if(z0, mark(z1), z2) → mark(if(z0, z1, z2))
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2))
proper(f(z0)) → f(proper(z0))
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2))
proper(c) → ok(c)
proper(true) → ok(true)
proper(false) → ok(false)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1))
F(mark(z0)) → c7(F(z0))
F(ok(z0)) → c8(F(z0))
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2))
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))
PROPER(f(z0)) → c12(F(proper(z0)), PROPER(z0))
PROPER(if(z0, z1, z2)) → c13(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
TOP(mark(z0)) → c17(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c18(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c1(IF(z0, c, f(true)))
ACTIVE(f(f(z0))) → c4(F(mark(if(z0, c, f(true)))), ACTIVE(f(z0)))
ACTIVE(f(if(true, z0, z1))) → c4(F(mark(z0)), ACTIVE(if(true, z0, z1)))
ACTIVE(f(if(false, z0, z1))) → c4(F(mark(z1)), ACTIVE(if(false, z0, z1)))
ACTIVE(f(f(z0))) → c4(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(z0, active(z1), z2)), ACTIVE(if(z0, z1, z2)))

S tuples:

ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1))
F(mark(z0)) → c7(F(z0))
F(ok(z0)) → c8(F(z0))
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2))
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))
PROPER(f(z0)) → c12(F(proper(z0)), PROPER(z0))
PROPER(if(z0, z1, z2)) → c13(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
TOP(ok(z0)) → c18(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c1(IF(z0, c, f(true)))
ACTIVE(f(f(z0))) → c4(F(mark(if(z0, c, f(true)))), ACTIVE(f(z0)))
ACTIVE(f(if(true, z0, z1))) → c4(F(mark(z0)), ACTIVE(if(true, z0, z1)))
ACTIVE(f(if(false, z0, z1))) → c4(F(mark(z1)), ACTIVE(if(false, z0, z1)))
ACTIVE(f(f(z0))) → c4(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(z0, active(z1), z2)), ACTIVE(if(z0, z1, z2)))

K tuples:

TOP(mark(z0)) → c17(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, f, if, proper, top

Defined Pair Symbols:

ACTIVE, F, IF, PROPER, TOP

Compound Symbols:

c5, c6, c7, c8, c9, c10, c11, c12, c13, c17, c18, c1, c4

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

Use narrowing to replace PROPER(f(z0)) → c12(F(proper(z0)), PROPER(z0)) by

PROPER(f(f(z0))) → c12(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(if(z0, z1, z2))) → c12(F(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
PROPER(f(c)) → c12(F(ok(c)), PROPER(c))
PROPER(f(true)) → c12(F(ok(true)), PROPER(true))
PROPER(f(false)) → c12(F(ok(false)), PROPER(false))
PROPER(f(x0)) → c12

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

Tuples:

ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1))
F(mark(z0)) → c7(F(z0))
F(ok(z0)) → c8(F(z0))
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2))
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))
PROPER(if(z0, z1, z2)) → c13(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
TOP(mark(z0)) → c17(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c18(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c1(IF(z0, c, f(true)))
ACTIVE(f(f(z0))) → c4(F(mark(if(z0, c, f(true)))), ACTIVE(f(z0)))
ACTIVE(f(if(true, z0, z1))) → c4(F(mark(z0)), ACTIVE(if(true, z0, z1)))
ACTIVE(f(if(false, z0, z1))) → c4(F(mark(z1)), ACTIVE(if(false, z0, z1)))
ACTIVE(f(f(z0))) → c4(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(z0, active(z1), z2)), ACTIVE(if(z0, z1, z2)))
PROPER(f(f(z0))) → c12(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(if(z0, z1, z2))) → c12(F(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
PROPER(f(c)) → c12(F(ok(c)), PROPER(c))
PROPER(f(true)) → c12(F(ok(true)), PROPER(true))
PROPER(f(false)) → c12(F(ok(false)), PROPER(false))
PROPER(f(x0)) → c12

S tuples:

ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1))
F(mark(z0)) → c7(F(z0))
F(ok(z0)) → c8(F(z0))
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2))
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))
PROPER(if(z0, z1, z2)) → c13(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
TOP(ok(z0)) → c18(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c1(IF(z0, c, f(true)))
ACTIVE(f(f(z0))) → c4(F(mark(if(z0, c, f(true)))), ACTIVE(f(z0)))
ACTIVE(f(if(true, z0, z1))) → c4(F(mark(z0)), ACTIVE(if(true, z0, z1)))
ACTIVE(f(if(false, z0, z1))) → c4(F(mark(z1)), ACTIVE(if(false, z0, z1)))
ACTIVE(f(f(z0))) → c4(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(z0, active(z1), z2)), ACTIVE(if(z0, z1, z2)))
PROPER(f(f(z0))) → c12(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(if(z0, z1, z2))) → c12(F(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
PROPER(f(c)) → c12(F(ok(c)), PROPER(c))
PROPER(f(true)) → c12(F(ok(true)), PROPER(true))
PROPER(f(false)) → c12(F(ok(false)), PROPER(false))
PROPER(f(x0)) → c12

K tuples:

TOP(mark(z0)) → c17(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, f, if, proper, top

Defined Pair Symbols:

ACTIVE, F, IF, PROPER, TOP

Compound Symbols:

c5, c6, c7, c8, c9, c10, c11, c13, c17, c18, c1, c4, c12, c12

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

Removed 1 trailing nodes:

PROPER(f(x0)) → c12

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

Tuples:

ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1))
F(mark(z0)) → c7(F(z0))
F(ok(z0)) → c8(F(z0))
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2))
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))
PROPER(if(z0, z1, z2)) → c13(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
TOP(mark(z0)) → c17(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c18(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c1(IF(z0, c, f(true)))
ACTIVE(f(f(z0))) → c4(F(mark(if(z0, c, f(true)))), ACTIVE(f(z0)))
ACTIVE(f(if(true, z0, z1))) → c4(F(mark(z0)), ACTIVE(if(true, z0, z1)))
ACTIVE(f(if(false, z0, z1))) → c4(F(mark(z1)), ACTIVE(if(false, z0, z1)))
ACTIVE(f(f(z0))) → c4(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(z0, active(z1), z2)), ACTIVE(if(z0, z1, z2)))
PROPER(f(f(z0))) → c12(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(if(z0, z1, z2))) → c12(F(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
PROPER(f(c)) → c12(F(ok(c)), PROPER(c))
PROPER(f(true)) → c12(F(ok(true)), PROPER(true))
PROPER(f(false)) → c12(F(ok(false)), PROPER(false))

S tuples:

ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1))
F(mark(z0)) → c7(F(z0))
F(ok(z0)) → c8(F(z0))
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2))
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))
PROPER(if(z0, z1, z2)) → c13(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
TOP(ok(z0)) → c18(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c1(IF(z0, c, f(true)))
ACTIVE(f(f(z0))) → c4(F(mark(if(z0, c, f(true)))), ACTIVE(f(z0)))
ACTIVE(f(if(true, z0, z1))) → c4(F(mark(z0)), ACTIVE(if(true, z0, z1)))
ACTIVE(f(if(false, z0, z1))) → c4(F(mark(z1)), ACTIVE(if(false, z0, z1)))
ACTIVE(f(f(z0))) → c4(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(z0, active(z1), z2)), ACTIVE(if(z0, z1, z2)))
PROPER(f(f(z0))) → c12(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(if(z0, z1, z2))) → c12(F(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
PROPER(f(c)) → c12(F(ok(c)), PROPER(c))
PROPER(f(true)) → c12(F(ok(true)), PROPER(true))
PROPER(f(false)) → c12(F(ok(false)), PROPER(false))

K tuples:

TOP(mark(z0)) → c17(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, f, if, proper, top

Defined Pair Symbols:

ACTIVE, F, IF, PROPER, TOP

Compound Symbols:

c5, c6, c7, c8, c9, c10, c11, c13, c17, c18, c1, c4, c12

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

Removed 3 trailing tuple parts

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

Tuples:

ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1))
F(mark(z0)) → c7(F(z0))
F(ok(z0)) → c8(F(z0))
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2))
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))
PROPER(if(z0, z1, z2)) → c13(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
TOP(mark(z0)) → c17(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c18(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c1(IF(z0, c, f(true)))
ACTIVE(f(f(z0))) → c4(F(mark(if(z0, c, f(true)))), ACTIVE(f(z0)))
ACTIVE(f(if(true, z0, z1))) → c4(F(mark(z0)), ACTIVE(if(true, z0, z1)))
ACTIVE(f(if(false, z0, z1))) → c4(F(mark(z1)), ACTIVE(if(false, z0, z1)))
ACTIVE(f(f(z0))) → c4(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(z0, active(z1), z2)), ACTIVE(if(z0, z1, z2)))
PROPER(f(f(z0))) → c12(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(if(z0, z1, z2))) → c12(F(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
PROPER(f(c)) → c12(F(ok(c)))
PROPER(f(true)) → c12(F(ok(true)))
PROPER(f(false)) → c12(F(ok(false)))

S tuples:

ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1))
F(mark(z0)) → c7(F(z0))
F(ok(z0)) → c8(F(z0))
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2))
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))
PROPER(if(z0, z1, z2)) → c13(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
TOP(ok(z0)) → c18(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c1(IF(z0, c, f(true)))
ACTIVE(f(f(z0))) → c4(F(mark(if(z0, c, f(true)))), ACTIVE(f(z0)))
ACTIVE(f(if(true, z0, z1))) → c4(F(mark(z0)), ACTIVE(if(true, z0, z1)))
ACTIVE(f(if(false, z0, z1))) → c4(F(mark(z1)), ACTIVE(if(false, z0, z1)))
ACTIVE(f(f(z0))) → c4(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(z0, active(z1), z2)), ACTIVE(if(z0, z1, z2)))
PROPER(f(f(z0))) → c12(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(if(z0, z1, z2))) → c12(F(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
PROPER(f(c)) → c12(F(ok(c)))
PROPER(f(true)) → c12(F(ok(true)))
PROPER(f(false)) → c12(F(ok(false)))

K tuples:

TOP(mark(z0)) → c17(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, f, if, proper, top

Defined Pair Symbols:

ACTIVE, F, IF, PROPER, TOP

Compound Symbols:

c5, c6, c7, c8, c9, c10, c11, c13, c17, c18, c1, c4, c12, c12

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

Use narrowing to replace PROPER(if(z0, z1, z2)) → c13(IF(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) by

PROPER(if(x0, x1, f(z0))) → c13(IF(proper(x0), proper(x1), f(proper(z0))), PROPER(x0), PROPER(x1), PROPER(f(z0)))
PROPER(if(x0, x1, if(z0, z1, z2))) → c13(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2)))
PROPER(if(x0, x1, c)) → c13(IF(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1), PROPER(c))
PROPER(if(x0, x1, true)) → c13(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1), PROPER(true))
PROPER(if(x0, x1, false)) → c13(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1), PROPER(false))
PROPER(if(x0, f(z0), x2)) → c13(IF(proper(x0), f(proper(z0)), proper(x2)), PROPER(x0), PROPER(f(z0)), PROPER(x2))
PROPER(if(x0, if(z0, z1, z2), x2)) → c13(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2))
PROPER(if(x0, c, x2)) → c13(IF(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(c), PROPER(x2))
PROPER(if(x0, true, x2)) → c13(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(true), PROPER(x2))
PROPER(if(x0, false, x2)) → c13(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(false), PROPER(x2))
PROPER(if(f(z0), x1, x2)) → c13(IF(f(proper(z0)), proper(x1), proper(x2)), PROPER(f(z0)), PROPER(x1), PROPER(x2))
PROPER(if(if(z0, z1, z2), x1, x2)) → c13(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(if(c, x1, x2)) → c13(IF(ok(c), proper(x1), proper(x2)), PROPER(c), PROPER(x1), PROPER(x2))
PROPER(if(true, x1, x2)) → c13(IF(ok(true), proper(x1), proper(x2)), PROPER(true), PROPER(x1), PROPER(x2))
PROPER(if(false, x1, x2)) → c13(IF(ok(false), proper(x1), proper(x2)), PROPER(false), PROPER(x1), PROPER(x2))
PROPER(if(x0, x1, x2)) → c13

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

Tuples:

ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1))
F(mark(z0)) → c7(F(z0))
F(ok(z0)) → c8(F(z0))
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2))
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))
TOP(mark(z0)) → c17(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c18(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c1(IF(z0, c, f(true)))
ACTIVE(f(f(z0))) → c4(F(mark(if(z0, c, f(true)))), ACTIVE(f(z0)))
ACTIVE(f(if(true, z0, z1))) → c4(F(mark(z0)), ACTIVE(if(true, z0, z1)))
ACTIVE(f(if(false, z0, z1))) → c4(F(mark(z1)), ACTIVE(if(false, z0, z1)))
ACTIVE(f(f(z0))) → c4(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(z0, active(z1), z2)), ACTIVE(if(z0, z1, z2)))
PROPER(f(f(z0))) → c12(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(if(z0, z1, z2))) → c12(F(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
PROPER(f(c)) → c12(F(ok(c)))
PROPER(f(true)) → c12(F(ok(true)))
PROPER(f(false)) → c12(F(ok(false)))
PROPER(if(x0, x1, f(z0))) → c13(IF(proper(x0), proper(x1), f(proper(z0))), PROPER(x0), PROPER(x1), PROPER(f(z0)))
PROPER(if(x0, x1, if(z0, z1, z2))) → c13(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2)))
PROPER(if(x0, x1, c)) → c13(IF(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1), PROPER(c))
PROPER(if(x0, x1, true)) → c13(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1), PROPER(true))
PROPER(if(x0, x1, false)) → c13(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1), PROPER(false))
PROPER(if(x0, f(z0), x2)) → c13(IF(proper(x0), f(proper(z0)), proper(x2)), PROPER(x0), PROPER(f(z0)), PROPER(x2))
PROPER(if(x0, if(z0, z1, z2), x2)) → c13(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2))
PROPER(if(x0, c, x2)) → c13(IF(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(c), PROPER(x2))
PROPER(if(x0, true, x2)) → c13(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(true), PROPER(x2))
PROPER(if(x0, false, x2)) → c13(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(false), PROPER(x2))
PROPER(if(f(z0), x1, x2)) → c13(IF(f(proper(z0)), proper(x1), proper(x2)), PROPER(f(z0)), PROPER(x1), PROPER(x2))
PROPER(if(if(z0, z1, z2), x1, x2)) → c13(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(if(c, x1, x2)) → c13(IF(ok(c), proper(x1), proper(x2)), PROPER(c), PROPER(x1), PROPER(x2))
PROPER(if(true, x1, x2)) → c13(IF(ok(true), proper(x1), proper(x2)), PROPER(true), PROPER(x1), PROPER(x2))
PROPER(if(false, x1, x2)) → c13(IF(ok(false), proper(x1), proper(x2)), PROPER(false), PROPER(x1), PROPER(x2))
PROPER(if(x0, x1, x2)) → c13

S tuples:

ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1))
F(mark(z0)) → c7(F(z0))
F(ok(z0)) → c8(F(z0))
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2))
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))
TOP(ok(z0)) → c18(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c1(IF(z0, c, f(true)))
ACTIVE(f(f(z0))) → c4(F(mark(if(z0, c, f(true)))), ACTIVE(f(z0)))
ACTIVE(f(if(true, z0, z1))) → c4(F(mark(z0)), ACTIVE(if(true, z0, z1)))
ACTIVE(f(if(false, z0, z1))) → c4(F(mark(z1)), ACTIVE(if(false, z0, z1)))
ACTIVE(f(f(z0))) → c4(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(z0, active(z1), z2)), ACTIVE(if(z0, z1, z2)))
PROPER(f(f(z0))) → c12(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(if(z0, z1, z2))) → c12(F(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
PROPER(f(c)) → c12(F(ok(c)))
PROPER(f(true)) → c12(F(ok(true)))
PROPER(f(false)) → c12(F(ok(false)))
PROPER(if(x0, x1, f(z0))) → c13(IF(proper(x0), proper(x1), f(proper(z0))), PROPER(x0), PROPER(x1), PROPER(f(z0)))
PROPER(if(x0, x1, if(z0, z1, z2))) → c13(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2)))
PROPER(if(x0, x1, c)) → c13(IF(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1), PROPER(c))
PROPER(if(x0, x1, true)) → c13(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1), PROPER(true))
PROPER(if(x0, x1, false)) → c13(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1), PROPER(false))
PROPER(if(x0, f(z0), x2)) → c13(IF(proper(x0), f(proper(z0)), proper(x2)), PROPER(x0), PROPER(f(z0)), PROPER(x2))
PROPER(if(x0, if(z0, z1, z2), x2)) → c13(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2))
PROPER(if(x0, c, x2)) → c13(IF(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(c), PROPER(x2))
PROPER(if(x0, true, x2)) → c13(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(true), PROPER(x2))
PROPER(if(x0, false, x2)) → c13(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(false), PROPER(x2))
PROPER(if(f(z0), x1, x2)) → c13(IF(f(proper(z0)), proper(x1), proper(x2)), PROPER(f(z0)), PROPER(x1), PROPER(x2))
PROPER(if(if(z0, z1, z2), x1, x2)) → c13(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(if(c, x1, x2)) → c13(IF(ok(c), proper(x1), proper(x2)), PROPER(c), PROPER(x1), PROPER(x2))
PROPER(if(true, x1, x2)) → c13(IF(ok(true), proper(x1), proper(x2)), PROPER(true), PROPER(x1), PROPER(x2))
PROPER(if(false, x1, x2)) → c13(IF(ok(false), proper(x1), proper(x2)), PROPER(false), PROPER(x1), PROPER(x2))
PROPER(if(x0, x1, x2)) → c13

K tuples:

TOP(mark(z0)) → c17(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, f, if, proper, top

Defined Pair Symbols:

ACTIVE, F, IF, TOP, PROPER

Compound Symbols:

c5, c6, c7, c8, c9, c10, c11, c17, c18, c1, c4, c12, c12, c13, c13

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

Removed 1 trailing nodes:

PROPER(if(x0, x1, x2)) → c13

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

Tuples:

ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1))
F(mark(z0)) → c7(F(z0))
F(ok(z0)) → c8(F(z0))
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2))
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))
TOP(mark(z0)) → c17(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c18(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c1(IF(z0, c, f(true)))
ACTIVE(f(f(z0))) → c4(F(mark(if(z0, c, f(true)))), ACTIVE(f(z0)))
ACTIVE(f(if(true, z0, z1))) → c4(F(mark(z0)), ACTIVE(if(true, z0, z1)))
ACTIVE(f(if(false, z0, z1))) → c4(F(mark(z1)), ACTIVE(if(false, z0, z1)))
ACTIVE(f(f(z0))) → c4(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(z0, active(z1), z2)), ACTIVE(if(z0, z1, z2)))
PROPER(f(f(z0))) → c12(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(if(z0, z1, z2))) → c12(F(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
PROPER(f(c)) → c12(F(ok(c)))
PROPER(f(true)) → c12(F(ok(true)))
PROPER(f(false)) → c12(F(ok(false)))
PROPER(if(x0, x1, f(z0))) → c13(IF(proper(x0), proper(x1), f(proper(z0))), PROPER(x0), PROPER(x1), PROPER(f(z0)))
PROPER(if(x0, x1, if(z0, z1, z2))) → c13(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2)))
PROPER(if(x0, x1, c)) → c13(IF(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1), PROPER(c))
PROPER(if(x0, x1, true)) → c13(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1), PROPER(true))
PROPER(if(x0, x1, false)) → c13(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1), PROPER(false))
PROPER(if(x0, f(z0), x2)) → c13(IF(proper(x0), f(proper(z0)), proper(x2)), PROPER(x0), PROPER(f(z0)), PROPER(x2))
PROPER(if(x0, if(z0, z1, z2), x2)) → c13(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2))
PROPER(if(x0, c, x2)) → c13(IF(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(c), PROPER(x2))
PROPER(if(x0, true, x2)) → c13(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(true), PROPER(x2))
PROPER(if(x0, false, x2)) → c13(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(false), PROPER(x2))
PROPER(if(f(z0), x1, x2)) → c13(IF(f(proper(z0)), proper(x1), proper(x2)), PROPER(f(z0)), PROPER(x1), PROPER(x2))
PROPER(if(if(z0, z1, z2), x1, x2)) → c13(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(if(c, x1, x2)) → c13(IF(ok(c), proper(x1), proper(x2)), PROPER(c), PROPER(x1), PROPER(x2))
PROPER(if(true, x1, x2)) → c13(IF(ok(true), proper(x1), proper(x2)), PROPER(true), PROPER(x1), PROPER(x2))
PROPER(if(false, x1, x2)) → c13(IF(ok(false), proper(x1), proper(x2)), PROPER(false), PROPER(x1), PROPER(x2))

S tuples:

ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1))
F(mark(z0)) → c7(F(z0))
F(ok(z0)) → c8(F(z0))
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2))
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))
TOP(ok(z0)) → c18(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c1(IF(z0, c, f(true)))
ACTIVE(f(f(z0))) → c4(F(mark(if(z0, c, f(true)))), ACTIVE(f(z0)))
ACTIVE(f(if(true, z0, z1))) → c4(F(mark(z0)), ACTIVE(if(true, z0, z1)))
ACTIVE(f(if(false, z0, z1))) → c4(F(mark(z1)), ACTIVE(if(false, z0, z1)))
ACTIVE(f(f(z0))) → c4(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(z0, active(z1), z2)), ACTIVE(if(z0, z1, z2)))
PROPER(f(f(z0))) → c12(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(if(z0, z1, z2))) → c12(F(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
PROPER(f(c)) → c12(F(ok(c)))
PROPER(f(true)) → c12(F(ok(true)))
PROPER(f(false)) → c12(F(ok(false)))
PROPER(if(x0, x1, f(z0))) → c13(IF(proper(x0), proper(x1), f(proper(z0))), PROPER(x0), PROPER(x1), PROPER(f(z0)))
PROPER(if(x0, x1, if(z0, z1, z2))) → c13(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2)))
PROPER(if(x0, x1, c)) → c13(IF(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1), PROPER(c))
PROPER(if(x0, x1, true)) → c13(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1), PROPER(true))
PROPER(if(x0, x1, false)) → c13(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1), PROPER(false))
PROPER(if(x0, f(z0), x2)) → c13(IF(proper(x0), f(proper(z0)), proper(x2)), PROPER(x0), PROPER(f(z0)), PROPER(x2))
PROPER(if(x0, if(z0, z1, z2), x2)) → c13(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2))
PROPER(if(x0, c, x2)) → c13(IF(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(c), PROPER(x2))
PROPER(if(x0, true, x2)) → c13(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(true), PROPER(x2))
PROPER(if(x0, false, x2)) → c13(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(false), PROPER(x2))
PROPER(if(f(z0), x1, x2)) → c13(IF(f(proper(z0)), proper(x1), proper(x2)), PROPER(f(z0)), PROPER(x1), PROPER(x2))
PROPER(if(if(z0, z1, z2), x1, x2)) → c13(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(if(c, x1, x2)) → c13(IF(ok(c), proper(x1), proper(x2)), PROPER(c), PROPER(x1), PROPER(x2))
PROPER(if(true, x1, x2)) → c13(IF(ok(true), proper(x1), proper(x2)), PROPER(true), PROPER(x1), PROPER(x2))
PROPER(if(false, x1, x2)) → c13(IF(ok(false), proper(x1), proper(x2)), PROPER(false), PROPER(x1), PROPER(x2))

K tuples:

TOP(mark(z0)) → c17(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, f, if, proper, top

Defined Pair Symbols:

ACTIVE, F, IF, TOP, PROPER

Compound Symbols:

c5, c6, c7, c8, c9, c10, c11, c17, c18, c1, c4, c12, c12, c13

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

Removed 9 trailing tuple parts

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

Tuples:

ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1))
F(mark(z0)) → c7(F(z0))
F(ok(z0)) → c8(F(z0))
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2))
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))
TOP(mark(z0)) → c17(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c18(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c1(IF(z0, c, f(true)))
ACTIVE(f(f(z0))) → c4(F(mark(if(z0, c, f(true)))), ACTIVE(f(z0)))
ACTIVE(f(if(true, z0, z1))) → c4(F(mark(z0)), ACTIVE(if(true, z0, z1)))
ACTIVE(f(if(false, z0, z1))) → c4(F(mark(z1)), ACTIVE(if(false, z0, z1)))
ACTIVE(f(f(z0))) → c4(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(z0, active(z1), z2)), ACTIVE(if(z0, z1, z2)))
PROPER(f(f(z0))) → c12(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(if(z0, z1, z2))) → c12(F(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
PROPER(f(c)) → c12(F(ok(c)))
PROPER(f(true)) → c12(F(ok(true)))
PROPER(f(false)) → c12(F(ok(false)))
PROPER(if(x0, x1, f(z0))) → c13(IF(proper(x0), proper(x1), f(proper(z0))), PROPER(x0), PROPER(x1), PROPER(f(z0)))
PROPER(if(x0, x1, if(z0, z1, z2))) → c13(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2)))
PROPER(if(x0, f(z0), x2)) → c13(IF(proper(x0), f(proper(z0)), proper(x2)), PROPER(x0), PROPER(f(z0)), PROPER(x2))
PROPER(if(x0, if(z0, z1, z2), x2)) → c13(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2))
PROPER(if(f(z0), x1, x2)) → c13(IF(f(proper(z0)), proper(x1), proper(x2)), PROPER(f(z0)), PROPER(x1), PROPER(x2))
PROPER(if(if(z0, z1, z2), x1, x2)) → c13(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(if(x0, x1, c)) → c13(IF(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, true)) → c13(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, false)) → c13(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1))
PROPER(if(x0, c, x2)) → c13(IF(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, true, x2)) → c13(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, false, x2)) → c13(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(c, x1, x2)) → c13(IF(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(true, x1, x2)) → c13(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(false, x1, x2)) → c13(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))

S tuples:

ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1))
F(mark(z0)) → c7(F(z0))
F(ok(z0)) → c8(F(z0))
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2))
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))
TOP(ok(z0)) → c18(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c1(IF(z0, c, f(true)))
ACTIVE(f(f(z0))) → c4(F(mark(if(z0, c, f(true)))), ACTIVE(f(z0)))
ACTIVE(f(if(true, z0, z1))) → c4(F(mark(z0)), ACTIVE(if(true, z0, z1)))
ACTIVE(f(if(false, z0, z1))) → c4(F(mark(z1)), ACTIVE(if(false, z0, z1)))
ACTIVE(f(f(z0))) → c4(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(z0, active(z1), z2)), ACTIVE(if(z0, z1, z2)))
PROPER(f(f(z0))) → c12(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(if(z0, z1, z2))) → c12(F(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
PROPER(f(c)) → c12(F(ok(c)))
PROPER(f(true)) → c12(F(ok(true)))
PROPER(f(false)) → c12(F(ok(false)))
PROPER(if(x0, x1, f(z0))) → c13(IF(proper(x0), proper(x1), f(proper(z0))), PROPER(x0), PROPER(x1), PROPER(f(z0)))
PROPER(if(x0, x1, if(z0, z1, z2))) → c13(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2)))
PROPER(if(x0, f(z0), x2)) → c13(IF(proper(x0), f(proper(z0)), proper(x2)), PROPER(x0), PROPER(f(z0)), PROPER(x2))
PROPER(if(x0, if(z0, z1, z2), x2)) → c13(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2))
PROPER(if(f(z0), x1, x2)) → c13(IF(f(proper(z0)), proper(x1), proper(x2)), PROPER(f(z0)), PROPER(x1), PROPER(x2))
PROPER(if(if(z0, z1, z2), x1, x2)) → c13(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(if(x0, x1, c)) → c13(IF(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, true)) → c13(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, false)) → c13(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1))
PROPER(if(x0, c, x2)) → c13(IF(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, true, x2)) → c13(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, false, x2)) → c13(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(c, x1, x2)) → c13(IF(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(true, x1, x2)) → c13(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(false, x1, x2)) → c13(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))

K tuples:

TOP(mark(z0)) → c17(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, f, if, proper, top

Defined Pair Symbols:

ACTIVE, F, IF, TOP, PROPER

Compound Symbols:

c5, c6, c7, c8, c9, c10, c11, c17, c18, c1, c4, c12, c12, c13, c13

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

Use narrowing to replace TOP(mark(z0)) → c17(TOP(proper(z0)), PROPER(z0)) by

TOP(mark(f(z0))) → c17(TOP(f(proper(z0))), PROPER(f(z0)))
TOP(mark(if(z0, z1, z2))) → c17(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
TOP(mark(c)) → c17(TOP(ok(c)), PROPER(c))
TOP(mark(true)) → c17(TOP(ok(true)), PROPER(true))
TOP(mark(false)) → c17(TOP(ok(false)), PROPER(false))
TOP(mark(x0)) → c17

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

Tuples:

ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1))
F(mark(z0)) → c7(F(z0))
F(ok(z0)) → c8(F(z0))
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2))
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))
TOP(ok(z0)) → c18(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c1(IF(z0, c, f(true)))
ACTIVE(f(f(z0))) → c4(F(mark(if(z0, c, f(true)))), ACTIVE(f(z0)))
ACTIVE(f(if(true, z0, z1))) → c4(F(mark(z0)), ACTIVE(if(true, z0, z1)))
ACTIVE(f(if(false, z0, z1))) → c4(F(mark(z1)), ACTIVE(if(false, z0, z1)))
ACTIVE(f(f(z0))) → c4(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(z0, active(z1), z2)), ACTIVE(if(z0, z1, z2)))
PROPER(f(f(z0))) → c12(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(if(z0, z1, z2))) → c12(F(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
PROPER(f(c)) → c12(F(ok(c)))
PROPER(f(true)) → c12(F(ok(true)))
PROPER(f(false)) → c12(F(ok(false)))
PROPER(if(x0, x1, f(z0))) → c13(IF(proper(x0), proper(x1), f(proper(z0))), PROPER(x0), PROPER(x1), PROPER(f(z0)))
PROPER(if(x0, x1, if(z0, z1, z2))) → c13(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2)))
PROPER(if(x0, f(z0), x2)) → c13(IF(proper(x0), f(proper(z0)), proper(x2)), PROPER(x0), PROPER(f(z0)), PROPER(x2))
PROPER(if(x0, if(z0, z1, z2), x2)) → c13(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2))
PROPER(if(f(z0), x1, x2)) → c13(IF(f(proper(z0)), proper(x1), proper(x2)), PROPER(f(z0)), PROPER(x1), PROPER(x2))
PROPER(if(if(z0, z1, z2), x1, x2)) → c13(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(if(x0, x1, c)) → c13(IF(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, true)) → c13(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, false)) → c13(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1))
PROPER(if(x0, c, x2)) → c13(IF(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, true, x2)) → c13(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, false, x2)) → c13(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(c, x1, x2)) → c13(IF(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(true, x1, x2)) → c13(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(false, x1, x2)) → c13(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(f(z0))) → c17(TOP(f(proper(z0))), PROPER(f(z0)))
TOP(mark(if(z0, z1, z2))) → c17(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
TOP(mark(c)) → c17(TOP(ok(c)), PROPER(c))
TOP(mark(true)) → c17(TOP(ok(true)), PROPER(true))
TOP(mark(false)) → c17(TOP(ok(false)), PROPER(false))
TOP(mark(x0)) → c17

S tuples:

ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1))
F(mark(z0)) → c7(F(z0))
F(ok(z0)) → c8(F(z0))
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2))
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))
TOP(ok(z0)) → c18(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c1(IF(z0, c, f(true)))
ACTIVE(f(f(z0))) → c4(F(mark(if(z0, c, f(true)))), ACTIVE(f(z0)))
ACTIVE(f(if(true, z0, z1))) → c4(F(mark(z0)), ACTIVE(if(true, z0, z1)))
ACTIVE(f(if(false, z0, z1))) → c4(F(mark(z1)), ACTIVE(if(false, z0, z1)))
ACTIVE(f(f(z0))) → c4(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(z0, active(z1), z2)), ACTIVE(if(z0, z1, z2)))
PROPER(f(f(z0))) → c12(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(if(z0, z1, z2))) → c12(F(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
PROPER(f(c)) → c12(F(ok(c)))
PROPER(f(true)) → c12(F(ok(true)))
PROPER(f(false)) → c12(F(ok(false)))
PROPER(if(x0, x1, f(z0))) → c13(IF(proper(x0), proper(x1), f(proper(z0))), PROPER(x0), PROPER(x1), PROPER(f(z0)))
PROPER(if(x0, x1, if(z0, z1, z2))) → c13(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2)))
PROPER(if(x0, f(z0), x2)) → c13(IF(proper(x0), f(proper(z0)), proper(x2)), PROPER(x0), PROPER(f(z0)), PROPER(x2))
PROPER(if(x0, if(z0, z1, z2), x2)) → c13(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2))
PROPER(if(f(z0), x1, x2)) → c13(IF(f(proper(z0)), proper(x1), proper(x2)), PROPER(f(z0)), PROPER(x1), PROPER(x2))
PROPER(if(if(z0, z1, z2), x1, x2)) → c13(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(if(x0, x1, c)) → c13(IF(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, true)) → c13(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, false)) → c13(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1))
PROPER(if(x0, c, x2)) → c13(IF(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, true, x2)) → c13(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, false, x2)) → c13(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(c, x1, x2)) → c13(IF(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(true, x1, x2)) → c13(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(false, x1, x2)) → c13(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))

K tuples:

TOP(mark(z0)) → c17(TOP(proper(z0)), PROPER(z0))

Defined Rule Symbols:

active, f, if, proper, top

Defined Pair Symbols:

ACTIVE, F, IF, TOP, PROPER

Compound Symbols:

c5, c6, c7, c8, c9, c10, c11, c18, c1, c4, c12, c12, c13, c13, c17, c17

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

Removed 1 trailing nodes:

TOP(mark(x0)) → c17

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

Tuples:

ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1))
F(mark(z0)) → c7(F(z0))
F(ok(z0)) → c8(F(z0))
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2))
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))
TOP(ok(z0)) → c18(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c1(IF(z0, c, f(true)))
ACTIVE(f(f(z0))) → c4(F(mark(if(z0, c, f(true)))), ACTIVE(f(z0)))
ACTIVE(f(if(true, z0, z1))) → c4(F(mark(z0)), ACTIVE(if(true, z0, z1)))
ACTIVE(f(if(false, z0, z1))) → c4(F(mark(z1)), ACTIVE(if(false, z0, z1)))
ACTIVE(f(f(z0))) → c4(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(z0, active(z1), z2)), ACTIVE(if(z0, z1, z2)))
PROPER(f(f(z0))) → c12(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(if(z0, z1, z2))) → c12(F(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
PROPER(f(c)) → c12(F(ok(c)))
PROPER(f(true)) → c12(F(ok(true)))
PROPER(f(false)) → c12(F(ok(false)))
PROPER(if(x0, x1, f(z0))) → c13(IF(proper(x0), proper(x1), f(proper(z0))), PROPER(x0), PROPER(x1), PROPER(f(z0)))
PROPER(if(x0, x1, if(z0, z1, z2))) → c13(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2)))
PROPER(if(x0, f(z0), x2)) → c13(IF(proper(x0), f(proper(z0)), proper(x2)), PROPER(x0), PROPER(f(z0)), PROPER(x2))
PROPER(if(x0, if(z0, z1, z2), x2)) → c13(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2))
PROPER(if(f(z0), x1, x2)) → c13(IF(f(proper(z0)), proper(x1), proper(x2)), PROPER(f(z0)), PROPER(x1), PROPER(x2))
PROPER(if(if(z0, z1, z2), x1, x2)) → c13(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(if(x0, x1, c)) → c13(IF(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, true)) → c13(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, false)) → c13(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1))
PROPER(if(x0, c, x2)) → c13(IF(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, true, x2)) → c13(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, false, x2)) → c13(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(c, x1, x2)) → c13(IF(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(true, x1, x2)) → c13(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(false, x1, x2)) → c13(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(f(z0))) → c17(TOP(f(proper(z0))), PROPER(f(z0)))
TOP(mark(if(z0, z1, z2))) → c17(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
TOP(mark(c)) → c17(TOP(ok(c)), PROPER(c))
TOP(mark(true)) → c17(TOP(ok(true)), PROPER(true))
TOP(mark(false)) → c17(TOP(ok(false)), PROPER(false))

S tuples:

ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1))
F(mark(z0)) → c7(F(z0))
F(ok(z0)) → c8(F(z0))
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2))
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))
TOP(ok(z0)) → c18(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c1(IF(z0, c, f(true)))
ACTIVE(f(f(z0))) → c4(F(mark(if(z0, c, f(true)))), ACTIVE(f(z0)))
ACTIVE(f(if(true, z0, z1))) → c4(F(mark(z0)), ACTIVE(if(true, z0, z1)))
ACTIVE(f(if(false, z0, z1))) → c4(F(mark(z1)), ACTIVE(if(false, z0, z1)))
ACTIVE(f(f(z0))) → c4(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(z0, active(z1), z2)), ACTIVE(if(z0, z1, z2)))
PROPER(f(f(z0))) → c12(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(if(z0, z1, z2))) → c12(F(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
PROPER(f(c)) → c12(F(ok(c)))
PROPER(f(true)) → c12(F(ok(true)))
PROPER(f(false)) → c12(F(ok(false)))
PROPER(if(x0, x1, f(z0))) → c13(IF(proper(x0), proper(x1), f(proper(z0))), PROPER(x0), PROPER(x1), PROPER(f(z0)))
PROPER(if(x0, x1, if(z0, z1, z2))) → c13(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2)))
PROPER(if(x0, f(z0), x2)) → c13(IF(proper(x0), f(proper(z0)), proper(x2)), PROPER(x0), PROPER(f(z0)), PROPER(x2))
PROPER(if(x0, if(z0, z1, z2), x2)) → c13(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2))
PROPER(if(f(z0), x1, x2)) → c13(IF(f(proper(z0)), proper(x1), proper(x2)), PROPER(f(z0)), PROPER(x1), PROPER(x2))
PROPER(if(if(z0, z1, z2), x1, x2)) → c13(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(if(x0, x1, c)) → c13(IF(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, true)) → c13(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, false)) → c13(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1))
PROPER(if(x0, c, x2)) → c13(IF(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, true, x2)) → c13(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, false, x2)) → c13(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(c, x1, x2)) → c13(IF(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(true, x1, x2)) → c13(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(false, x1, x2)) → c13(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))

K tuples:none
Defined Rule Symbols:

active, f, if, proper, top

Defined Pair Symbols:

ACTIVE, F, IF, TOP, PROPER

Compound Symbols:

c5, c6, c7, c8, c9, c10, c11, c18, c1, c4, c12, c12, c13, c13, c17

(27) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 3 trailing tuple parts

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

Tuples:

ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1))
F(mark(z0)) → c7(F(z0))
F(ok(z0)) → c8(F(z0))
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2))
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))
TOP(ok(z0)) → c18(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c1(IF(z0, c, f(true)))
ACTIVE(f(f(z0))) → c4(F(mark(if(z0, c, f(true)))), ACTIVE(f(z0)))
ACTIVE(f(if(true, z0, z1))) → c4(F(mark(z0)), ACTIVE(if(true, z0, z1)))
ACTIVE(f(if(false, z0, z1))) → c4(F(mark(z1)), ACTIVE(if(false, z0, z1)))
ACTIVE(f(f(z0))) → c4(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(z0, active(z1), z2)), ACTIVE(if(z0, z1, z2)))
PROPER(f(f(z0))) → c12(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(if(z0, z1, z2))) → c12(F(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
PROPER(f(c)) → c12(F(ok(c)))
PROPER(f(true)) → c12(F(ok(true)))
PROPER(f(false)) → c12(F(ok(false)))
PROPER(if(x0, x1, f(z0))) → c13(IF(proper(x0), proper(x1), f(proper(z0))), PROPER(x0), PROPER(x1), PROPER(f(z0)))
PROPER(if(x0, x1, if(z0, z1, z2))) → c13(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2)))
PROPER(if(x0, f(z0), x2)) → c13(IF(proper(x0), f(proper(z0)), proper(x2)), PROPER(x0), PROPER(f(z0)), PROPER(x2))
PROPER(if(x0, if(z0, z1, z2), x2)) → c13(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2))
PROPER(if(f(z0), x1, x2)) → c13(IF(f(proper(z0)), proper(x1), proper(x2)), PROPER(f(z0)), PROPER(x1), PROPER(x2))
PROPER(if(if(z0, z1, z2), x1, x2)) → c13(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(if(x0, x1, c)) → c13(IF(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, true)) → c13(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, false)) → c13(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1))
PROPER(if(x0, c, x2)) → c13(IF(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, true, x2)) → c13(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, false, x2)) → c13(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(c, x1, x2)) → c13(IF(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(true, x1, x2)) → c13(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(false, x1, x2)) → c13(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(f(z0))) → c17(TOP(f(proper(z0))), PROPER(f(z0)))
TOP(mark(if(z0, z1, z2))) → c17(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
TOP(mark(c)) → c17(TOP(ok(c)))
TOP(mark(true)) → c17(TOP(ok(true)))
TOP(mark(false)) → c17(TOP(ok(false)))

S tuples:

ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1))
F(mark(z0)) → c7(F(z0))
F(ok(z0)) → c8(F(z0))
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2))
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))
TOP(ok(z0)) → c18(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(z0)) → c1(IF(z0, c, f(true)))
ACTIVE(f(f(z0))) → c4(F(mark(if(z0, c, f(true)))), ACTIVE(f(z0)))
ACTIVE(f(if(true, z0, z1))) → c4(F(mark(z0)), ACTIVE(if(true, z0, z1)))
ACTIVE(f(if(false, z0, z1))) → c4(F(mark(z1)), ACTIVE(if(false, z0, z1)))
ACTIVE(f(f(z0))) → c4(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(z0, active(z1), z2)), ACTIVE(if(z0, z1, z2)))
PROPER(f(f(z0))) → c12(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(if(z0, z1, z2))) → c12(F(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
PROPER(f(c)) → c12(F(ok(c)))
PROPER(f(true)) → c12(F(ok(true)))
PROPER(f(false)) → c12(F(ok(false)))
PROPER(if(x0, x1, f(z0))) → c13(IF(proper(x0), proper(x1), f(proper(z0))), PROPER(x0), PROPER(x1), PROPER(f(z0)))
PROPER(if(x0, x1, if(z0, z1, z2))) → c13(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2)))
PROPER(if(x0, f(z0), x2)) → c13(IF(proper(x0), f(proper(z0)), proper(x2)), PROPER(x0), PROPER(f(z0)), PROPER(x2))
PROPER(if(x0, if(z0, z1, z2), x2)) → c13(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2))
PROPER(if(f(z0), x1, x2)) → c13(IF(f(proper(z0)), proper(x1), proper(x2)), PROPER(f(z0)), PROPER(x1), PROPER(x2))
PROPER(if(if(z0, z1, z2), x1, x2)) → c13(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(if(x0, x1, c)) → c13(IF(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, true)) → c13(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, false)) → c13(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1))
PROPER(if(x0, c, x2)) → c13(IF(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, true, x2)) → c13(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, false, x2)) → c13(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(c, x1, x2)) → c13(IF(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(true, x1, x2)) → c13(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(false, x1, x2)) → c13(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))

K tuples:none
Defined Rule Symbols:

active, f, if, proper, top

Defined Pair Symbols:

ACTIVE, F, IF, TOP, PROPER

Compound Symbols:

c5, c6, c7, c8, c9, c10, c11, c18, c1, c4, c12, c12, c13, c13, c17, c17

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

Use narrowing to replace TOP(ok(z0)) → c18(TOP(active(z0)), ACTIVE(z0)) by

TOP(ok(f(z0))) → c18(TOP(mark(if(z0, c, f(true)))), ACTIVE(f(z0)))
TOP(ok(if(true, z0, z1))) → c18(TOP(mark(z0)), ACTIVE(if(true, z0, z1)))
TOP(ok(if(false, z0, z1))) → c18(TOP(mark(z1)), ACTIVE(if(false, z0, z1)))
TOP(ok(f(z0))) → c18(TOP(f(active(z0))), ACTIVE(f(z0)))
TOP(ok(if(z0, z1, z2))) → c18(TOP(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
TOP(ok(if(z0, z1, z2))) → c18(TOP(if(z0, active(z1), z2)), ACTIVE(if(z0, z1, z2)))
TOP(ok(x0)) → c18

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

Tuples:

ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1))
F(mark(z0)) → c7(F(z0))
F(ok(z0)) → c8(F(z0))
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2))
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))
ACTIVE(f(z0)) → c1(IF(z0, c, f(true)))
ACTIVE(f(f(z0))) → c4(F(mark(if(z0, c, f(true)))), ACTIVE(f(z0)))
ACTIVE(f(if(true, z0, z1))) → c4(F(mark(z0)), ACTIVE(if(true, z0, z1)))
ACTIVE(f(if(false, z0, z1))) → c4(F(mark(z1)), ACTIVE(if(false, z0, z1)))
ACTIVE(f(f(z0))) → c4(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(z0, active(z1), z2)), ACTIVE(if(z0, z1, z2)))
PROPER(f(f(z0))) → c12(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(if(z0, z1, z2))) → c12(F(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
PROPER(f(c)) → c12(F(ok(c)))
PROPER(f(true)) → c12(F(ok(true)))
PROPER(f(false)) → c12(F(ok(false)))
PROPER(if(x0, x1, f(z0))) → c13(IF(proper(x0), proper(x1), f(proper(z0))), PROPER(x0), PROPER(x1), PROPER(f(z0)))
PROPER(if(x0, x1, if(z0, z1, z2))) → c13(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2)))
PROPER(if(x0, f(z0), x2)) → c13(IF(proper(x0), f(proper(z0)), proper(x2)), PROPER(x0), PROPER(f(z0)), PROPER(x2))
PROPER(if(x0, if(z0, z1, z2), x2)) → c13(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2))
PROPER(if(f(z0), x1, x2)) → c13(IF(f(proper(z0)), proper(x1), proper(x2)), PROPER(f(z0)), PROPER(x1), PROPER(x2))
PROPER(if(if(z0, z1, z2), x1, x2)) → c13(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(if(x0, x1, c)) → c13(IF(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, true)) → c13(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, false)) → c13(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1))
PROPER(if(x0, c, x2)) → c13(IF(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, true, x2)) → c13(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, false, x2)) → c13(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(c, x1, x2)) → c13(IF(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(true, x1, x2)) → c13(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(false, x1, x2)) → c13(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(f(z0))) → c17(TOP(f(proper(z0))), PROPER(f(z0)))
TOP(mark(if(z0, z1, z2))) → c17(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
TOP(mark(c)) → c17(TOP(ok(c)))
TOP(mark(true)) → c17(TOP(ok(true)))
TOP(mark(false)) → c17(TOP(ok(false)))
TOP(ok(f(z0))) → c18(TOP(mark(if(z0, c, f(true)))), ACTIVE(f(z0)))
TOP(ok(if(true, z0, z1))) → c18(TOP(mark(z0)), ACTIVE(if(true, z0, z1)))
TOP(ok(if(false, z0, z1))) → c18(TOP(mark(z1)), ACTIVE(if(false, z0, z1)))
TOP(ok(f(z0))) → c18(TOP(f(active(z0))), ACTIVE(f(z0)))
TOP(ok(if(z0, z1, z2))) → c18(TOP(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
TOP(ok(if(z0, z1, z2))) → c18(TOP(if(z0, active(z1), z2)), ACTIVE(if(z0, z1, z2)))
TOP(ok(x0)) → c18

S tuples:

ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1))
F(mark(z0)) → c7(F(z0))
F(ok(z0)) → c8(F(z0))
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2))
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))
ACTIVE(f(z0)) → c1(IF(z0, c, f(true)))
ACTIVE(f(f(z0))) → c4(F(mark(if(z0, c, f(true)))), ACTIVE(f(z0)))
ACTIVE(f(if(true, z0, z1))) → c4(F(mark(z0)), ACTIVE(if(true, z0, z1)))
ACTIVE(f(if(false, z0, z1))) → c4(F(mark(z1)), ACTIVE(if(false, z0, z1)))
ACTIVE(f(f(z0))) → c4(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(z0, active(z1), z2)), ACTIVE(if(z0, z1, z2)))
PROPER(f(f(z0))) → c12(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(if(z0, z1, z2))) → c12(F(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
PROPER(f(c)) → c12(F(ok(c)))
PROPER(f(true)) → c12(F(ok(true)))
PROPER(f(false)) → c12(F(ok(false)))
PROPER(if(x0, x1, f(z0))) → c13(IF(proper(x0), proper(x1), f(proper(z0))), PROPER(x0), PROPER(x1), PROPER(f(z0)))
PROPER(if(x0, x1, if(z0, z1, z2))) → c13(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2)))
PROPER(if(x0, f(z0), x2)) → c13(IF(proper(x0), f(proper(z0)), proper(x2)), PROPER(x0), PROPER(f(z0)), PROPER(x2))
PROPER(if(x0, if(z0, z1, z2), x2)) → c13(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2))
PROPER(if(f(z0), x1, x2)) → c13(IF(f(proper(z0)), proper(x1), proper(x2)), PROPER(f(z0)), PROPER(x1), PROPER(x2))
PROPER(if(if(z0, z1, z2), x1, x2)) → c13(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(if(x0, x1, c)) → c13(IF(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, true)) → c13(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, false)) → c13(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1))
PROPER(if(x0, c, x2)) → c13(IF(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, true, x2)) → c13(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, false, x2)) → c13(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(c, x1, x2)) → c13(IF(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(true, x1, x2)) → c13(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(false, x1, x2)) → c13(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(ok(f(z0))) → c18(TOP(mark(if(z0, c, f(true)))), ACTIVE(f(z0)))
TOP(ok(if(true, z0, z1))) → c18(TOP(mark(z0)), ACTIVE(if(true, z0, z1)))
TOP(ok(if(false, z0, z1))) → c18(TOP(mark(z1)), ACTIVE(if(false, z0, z1)))
TOP(ok(f(z0))) → c18(TOP(f(active(z0))), ACTIVE(f(z0)))
TOP(ok(if(z0, z1, z2))) → c18(TOP(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
TOP(ok(if(z0, z1, z2))) → c18(TOP(if(z0, active(z1), z2)), ACTIVE(if(z0, z1, z2)))
TOP(ok(x0)) → c18

K tuples:none
Defined Rule Symbols:

active, f, if, proper, top

Defined Pair Symbols:

ACTIVE, F, IF, PROPER, TOP

Compound Symbols:

c5, c6, c7, c8, c9, c10, c11, c1, c4, c12, c12, c13, c13, c17, c17, c18, c18

(31) CdtUnreachableProof (EQUIVALENT transformation)

The following tuples could be removed as they are not reachable from basic start terms:

ACTIVE(if(z0, z1, z2)) → c5(IF(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(if(z0, z1, z2)) → c6(IF(z0, active(z1), z2), ACTIVE(z1))
ACTIVE(f(z0)) → c1(IF(z0, c, f(true)))
ACTIVE(f(f(z0))) → c4(F(mark(if(z0, c, f(true)))), ACTIVE(f(z0)))
ACTIVE(f(if(true, z0, z1))) → c4(F(mark(z0)), ACTIVE(if(true, z0, z1)))
ACTIVE(f(if(false, z0, z1))) → c4(F(mark(z1)), ACTIVE(if(false, z0, z1)))
ACTIVE(f(f(z0))) → c4(F(f(active(z0))), ACTIVE(f(z0)))
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
ACTIVE(f(if(z0, z1, z2))) → c4(F(if(z0, active(z1), z2)), ACTIVE(if(z0, z1, z2)))
PROPER(f(f(z0))) → c12(F(f(proper(z0))), PROPER(f(z0)))
PROPER(f(if(z0, z1, z2))) → c12(F(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
PROPER(f(c)) → c12(F(ok(c)))
PROPER(f(true)) → c12(F(ok(true)))
PROPER(f(false)) → c12(F(ok(false)))
PROPER(if(x0, x1, f(z0))) → c13(IF(proper(x0), proper(x1), f(proper(z0))), PROPER(x0), PROPER(x1), PROPER(f(z0)))
PROPER(if(x0, x1, if(z0, z1, z2))) → c13(IF(proper(x0), proper(x1), if(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(if(z0, z1, z2)))
PROPER(if(x0, f(z0), x2)) → c13(IF(proper(x0), f(proper(z0)), proper(x2)), PROPER(x0), PROPER(f(z0)), PROPER(x2))
PROPER(if(x0, if(z0, z1, z2), x2)) → c13(IF(proper(x0), if(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(if(z0, z1, z2)), PROPER(x2))
PROPER(if(f(z0), x1, x2)) → c13(IF(f(proper(z0)), proper(x1), proper(x2)), PROPER(f(z0)), PROPER(x1), PROPER(x2))
PROPER(if(if(z0, z1, z2), x1, x2)) → c13(IF(if(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(if(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(if(x0, x1, c)) → c13(IF(proper(x0), proper(x1), ok(c)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, true)) → c13(IF(proper(x0), proper(x1), ok(true)), PROPER(x0), PROPER(x1))
PROPER(if(x0, x1, false)) → c13(IF(proper(x0), proper(x1), ok(false)), PROPER(x0), PROPER(x1))
PROPER(if(x0, c, x2)) → c13(IF(proper(x0), ok(c), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, true, x2)) → c13(IF(proper(x0), ok(true), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(x0, false, x2)) → c13(IF(proper(x0), ok(false), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(if(c, x1, x2)) → c13(IF(ok(c), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(true, x1, x2)) → c13(IF(ok(true), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(if(false, x1, x2)) → c13(IF(ok(false), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
TOP(mark(f(z0))) → c17(TOP(f(proper(z0))), PROPER(f(z0)))
TOP(mark(if(z0, z1, z2))) → c17(TOP(if(proper(z0), proper(z1), proper(z2))), PROPER(if(z0, z1, z2)))
TOP(ok(f(z0))) → c18(TOP(mark(if(z0, c, f(true)))), ACTIVE(f(z0)))
TOP(ok(if(true, z0, z1))) → c18(TOP(mark(z0)), ACTIVE(if(true, z0, z1)))
TOP(ok(if(false, z0, z1))) → c18(TOP(mark(z1)), ACTIVE(if(false, z0, z1)))
TOP(ok(f(z0))) → c18(TOP(f(active(z0))), ACTIVE(f(z0)))
TOP(ok(if(z0, z1, z2))) → c18(TOP(if(active(z0), z1, z2)), ACTIVE(if(z0, z1, z2)))
TOP(ok(if(z0, z1, z2))) → c18(TOP(if(z0, active(z1), z2)), ACTIVE(if(z0, z1, z2)))

(32) 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)
active(f(z0)) → f(active(z0))
active(if(z0, z1, z2)) → if(active(z0), z1, z2)
active(if(z0, z1, z2)) → if(z0, active(z1), z2)
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
if(mark(z0), z1, z2) → mark(if(z0, z1, z2))
if(z0, mark(z1), z2) → mark(if(z0, z1, z2))
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2))
proper(f(z0)) → f(proper(z0))
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2))
proper(c) → ok(c)
proper(true) → ok(true)
proper(false) → ok(false)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

F(mark(z0)) → c7(F(z0))
F(ok(z0)) → c8(F(z0))
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2))
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))
TOP(mark(c)) → c17(TOP(ok(c)))
TOP(mark(true)) → c17(TOP(ok(true)))
TOP(mark(false)) → c17(TOP(ok(false)))
TOP(ok(x0)) → c18

S tuples:

F(mark(z0)) → c7(F(z0))
F(ok(z0)) → c8(F(z0))
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2))
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))
TOP(ok(x0)) → c18

K tuples:none
Defined Rule Symbols:

active, f, if, proper, top

Defined Pair Symbols:

F, IF, TOP

Compound Symbols:

c7, c8, c9, c10, c11, c17, c18

(33) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 4 leading nodes:

TOP(mark(c)) → c17(TOP(ok(c)))
TOP(mark(true)) → c17(TOP(ok(true)))
TOP(mark(false)) → c17(TOP(ok(false)))
TOP(ok(x0)) → c18

(34) 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)
active(f(z0)) → f(active(z0))
active(if(z0, z1, z2)) → if(active(z0), z1, z2)
active(if(z0, z1, z2)) → if(z0, active(z1), z2)
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
if(mark(z0), z1, z2) → mark(if(z0, z1, z2))
if(z0, mark(z1), z2) → mark(if(z0, z1, z2))
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2))
proper(f(z0)) → f(proper(z0))
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2))
proper(c) → ok(c)
proper(true) → ok(true)
proper(false) → ok(false)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

F(mark(z0)) → c7(F(z0))
F(ok(z0)) → c8(F(z0))
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2))
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))

S tuples:

F(mark(z0)) → c7(F(z0))
F(ok(z0)) → c8(F(z0))
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2))
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))

K tuples:none
Defined Rule Symbols:

active, f, if, proper, top

Defined Pair Symbols:

F, IF

Compound Symbols:

c7, c8, c9, c10, c11

(35) 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.

IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))

We considered the (Usable) Rules:none
And the Tuples:

F(mark(z0)) → c7(F(z0))
F(ok(z0)) → c8(F(z0))
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2))
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))

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

POL(F(x₁)) = 0   
POL(IF(x₁, x₂, x₃)) = [2]x₃   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = [5]   
POL(ok(x₁)) = [5] + x₁   

(36) 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)
active(f(z0)) → f(active(z0))
active(if(z0, z1, z2)) → if(active(z0), z1, z2)
active(if(z0, z1, z2)) → if(z0, active(z1), z2)
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
if(mark(z0), z1, z2) → mark(if(z0, z1, z2))
if(z0, mark(z1), z2) → mark(if(z0, z1, z2))
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2))
proper(f(z0)) → f(proper(z0))
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2))
proper(c) → ok(c)
proper(true) → ok(true)
proper(false) → ok(false)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

F(mark(z0)) → c7(F(z0))
F(ok(z0)) → c8(F(z0))
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2))
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))

S tuples:

F(mark(z0)) → c7(F(z0))
F(ok(z0)) → c8(F(z0))
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2))
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2))

K tuples:

IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))

Defined Rule Symbols:

active, f, if, proper, top

Defined Pair Symbols:

F, IF

Compound Symbols:

c7, c8, c9, c10, c11

(37) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

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

F(mark(z0)) → c7(F(z0))
F(ok(z0)) → c8(F(z0))

We considered the (Usable) Rules:none
And the Tuples:

F(mark(z0)) → c7(F(z0))
F(ok(z0)) → c8(F(z0))
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2))
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))

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

POL(F(x₁)) = [2]x₁²   
POL(IF(x₁, x₂, x₃)) = [3]x₃² + [2]x₂·x₃   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = [1] + x₁   
POL(ok(x₁)) = [1] + x₁   

(38) 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)
active(f(z0)) → f(active(z0))
active(if(z0, z1, z2)) → if(active(z0), z1, z2)
active(if(z0, z1, z2)) → if(z0, active(z1), z2)
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
if(mark(z0), z1, z2) → mark(if(z0, z1, z2))
if(z0, mark(z1), z2) → mark(if(z0, z1, z2))
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2))
proper(f(z0)) → f(proper(z0))
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2))
proper(c) → ok(c)
proper(true) → ok(true)
proper(false) → ok(false)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

F(mark(z0)) → c7(F(z0))
F(ok(z0)) → c8(F(z0))
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2))
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))

S tuples:

IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2))
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2))

K tuples:

IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))
F(mark(z0)) → c7(F(z0))
F(ok(z0)) → c8(F(z0))

Defined Rule Symbols:

active, f, if, proper, top

Defined Pair Symbols:

F, IF

Compound Symbols:

c7, c8, c9, c10, c11

(39) 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.

IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2))

We considered the (Usable) Rules:none
And the Tuples:

F(mark(z0)) → c7(F(z0))
F(ok(z0)) → c8(F(z0))
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2))
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))

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

POL(F(x₁)) = 0   
POL(IF(x₁, x₂, x₃)) = x₂   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = [1] + x₁   
POL(ok(x₁)) = x₁   

(40) 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)
active(f(z0)) → f(active(z0))
active(if(z0, z1, z2)) → if(active(z0), z1, z2)
active(if(z0, z1, z2)) → if(z0, active(z1), z2)
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
if(mark(z0), z1, z2) → mark(if(z0, z1, z2))
if(z0, mark(z1), z2) → mark(if(z0, z1, z2))
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2))
proper(f(z0)) → f(proper(z0))
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2))
proper(c) → ok(c)
proper(true) → ok(true)
proper(false) → ok(false)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

F(mark(z0)) → c7(F(z0))
F(ok(z0)) → c8(F(z0))
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2))
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))

S tuples:

IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2))

K tuples:

IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))
F(mark(z0)) → c7(F(z0))
F(ok(z0)) → c8(F(z0))
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2))

Defined Rule Symbols:

active, f, if, proper, top

Defined Pair Symbols:

F, IF

Compound Symbols:

c7, c8, c9, c10, c11

(41) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

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

IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2))

We considered the (Usable) Rules:none
And the Tuples:

F(mark(z0)) → c7(F(z0))
F(ok(z0)) → c8(F(z0))
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2))
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))

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

POL(F(x₁)) = x₁ + [3]x₁²   
POL(IF(x₁, x₂, x₃)) = x₁ + [2]x₃ + [2]x₂·x₃ + [3]x₁·x₃   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = [1] + x₁   
POL(ok(x₁)) = [3] + x₁   

(42) 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)
active(f(z0)) → f(active(z0))
active(if(z0, z1, z2)) → if(active(z0), z1, z2)
active(if(z0, z1, z2)) → if(z0, active(z1), z2)
f(mark(z0)) → mark(f(z0))
f(ok(z0)) → ok(f(z0))
if(mark(z0), z1, z2) → mark(if(z0, z1, z2))
if(z0, mark(z1), z2) → mark(if(z0, z1, z2))
if(ok(z0), ok(z1), ok(z2)) → ok(if(z0, z1, z2))
proper(f(z0)) → f(proper(z0))
proper(if(z0, z1, z2)) → if(proper(z0), proper(z1), proper(z2))
proper(c) → ok(c)
proper(true) → ok(true)
proper(false) → ok(false)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

F(mark(z0)) → c7(F(z0))
F(ok(z0)) → c8(F(z0))
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2))
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2))
IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))

S tuples:none
K tuples:

IF(ok(z0), ok(z1), ok(z2)) → c11(IF(z0, z1, z2))
F(mark(z0)) → c7(F(z0))
F(ok(z0)) → c8(F(z0))
IF(z0, mark(z1), z2) → c10(IF(z0, z1, z2))
IF(mark(z0), z1, z2) → c9(IF(z0, z1, z2))

Defined Rule Symbols:

active, f, if, proper, top

Defined Pair Symbols:

F, IF

Compound Symbols:

c7, c8, c9, c10, c11

(43) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(44) BOUNDS(O(1), O(1))