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

0 CpxTRS
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
↳2 CdtProblem
↳3 CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication)
↳4 CdtProblem
↳5 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
↳6 CdtProblem
↳7 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
↳8 CdtProblem
↳9 CdtGraphRemoveTrailingProof (BOTH BOUNDS(ID, ID))
↳10 CdtProblem
↳11 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
↳12 CdtProblem
↳13 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
↳14 CdtProblem
↳15 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
↳16 CdtProblem
↳17 CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication)
↳18 CdtProblem
↳19 CdtGraphRemoveTrailingProof (BOTH BOUNDS(ID, ID))
↳20 CdtProblem
↳21 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
↳22 CdtProblem
↳23 CdtGraphRemoveTrailingProof (BOTH BOUNDS(ID, ID))
↳24 CdtProblem
↳25 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
↳26 CdtProblem
↳27 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
↳28 CdtProblem
↳29 CdtNarrowingProof (BOTH BOUNDS(ID, ID))
↳30 CdtProblem
↳31 CdtUnreachableProof (⇔)
↳32 CdtProblem
↳33 CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication)
↳34 CdtProblem
↳35 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
↳36 CdtProblem
↳37 SIsEmptyProof (⇔)
↳38 BOUNDS(O(1), O(1))

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

active(f(X, X)) → mark(f(a, b))
active(b) → mark(a)
active(f(X1, X2)) → f(active(X1), X2)
f(mark(X1), X2) → mark(f(X1, X2))
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(a) → ok(a)
proper(b) → ok(b)
f(ok(X1), ok(X2)) → ok(f(X1, X2))
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, z0)) → mark(f(a, b))
active(b) → mark(a)
active(f(z0, z1)) → f(active(z0), z1)
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(a) → ok(a)
proper(b) → ok(b)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(z0, z0)) → c(F(a, b))
ACTIVE(f(z0, z1)) → c2(F(active(z0), z1), ACTIVE(z0))
F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
PROPER(f(z0, z1)) → c5(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
S tuples:

ACTIVE(f(z0, z0)) → c(F(a, b))
ACTIVE(f(z0, z1)) → c2(F(active(z0), z1), ACTIVE(z0))
F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
PROPER(f(z0, z1)) → c5(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
K tuples:none
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

ACTIVE, F, PROPER, TOP

Compound Symbols:

c, c2, c3, c4, c5, c8, c9

(3) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 1 of 7 dangling nodes:

ACTIVE(f(z0, z0)) → c(F(a, b))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, z0)) → mark(f(a, b))
active(b) → mark(a)
active(f(z0, z1)) → f(active(z0), z1)
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(a) → ok(a)
proper(b) → ok(b)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

ACTIVE(f(z0, z1)) → c2(F(active(z0), z1), ACTIVE(z0))
F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
PROPER(f(z0, z1)) → c5(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
S tuples:

ACTIVE(f(z0, z1)) → c2(F(active(z0), z1), ACTIVE(z0))
F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
PROPER(f(z0, z1)) → c5(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
K tuples:none
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

ACTIVE, F, PROPER, TOP

Compound Symbols:

c2, c3, c4, c5, c8, c9

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

Use narrowing to replace ACTIVE(f(z0, z1)) → c2(F(active(z0), z1), ACTIVE(z0)) by

ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(b, x1)) → c2(F(mark(a), x1), ACTIVE(b))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, z0)) → mark(f(a, b))
active(b) → mark(a)
active(f(z0, z1)) → f(active(z0), z1)
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(a) → ok(a)
proper(b) → ok(b)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
PROPER(f(z0, z1)) → c5(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(b, x1)) → c2(F(mark(a), x1), ACTIVE(b))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
S tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
PROPER(f(z0, z1)) → c5(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(b, x1)) → c2(F(mark(a), x1), ACTIVE(b))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, PROPER, TOP, ACTIVE

Compound Symbols:

c3, c4, c5, c8, c9, c2

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

Split RHS of tuples not part of any SCC

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, z0)) → mark(f(a, b))
active(b) → mark(a)
active(f(z0, z1)) → f(active(z0), z1)
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(a) → ok(a)
proper(b) → ok(b)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
PROPER(f(z0, z1)) → c5(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(b, x1)) → c(F(mark(a), x1))
ACTIVE(f(b, x1)) → c(ACTIVE(b))
S tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
PROPER(f(z0, z1)) → c5(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(b, x1)) → c(F(mark(a), x1))
ACTIVE(f(b, x1)) → c(ACTIVE(b))
K tuples:none
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, PROPER, TOP, ACTIVE

Compound Symbols:

c3, c4, c5, c8, c9, c2, c

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

Removed 1 trailing tuple parts

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, z0)) → mark(f(a, b))
active(b) → mark(a)
active(f(z0, z1)) → f(active(z0), z1)
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(a) → ok(a)
proper(b) → ok(b)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
PROPER(f(z0, z1)) → c5(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(b, x1)) → c(F(mark(a), x1))
ACTIVE(f(b, x1)) → c
S tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
PROPER(f(z0, z1)) → c5(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(b, x1)) → c(F(mark(a), x1))
ACTIVE(f(b, x1)) → c
K tuples:none
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, PROPER, TOP, ACTIVE

Compound Symbols:

c3, c4, c5, c8, c9, c2, c, c

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

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

ACTIVE(f(b, x1)) → c(F(mark(a), x1))
We considered the (Usable) Rules:

active(f(z0, z0)) → mark(f(a, b))
active(b) → mark(a)
active(f(z0, z1)) → f(active(z0), z1)
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(a) → ok(a)
proper(b) → ok(b)
And the Tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
PROPER(f(z0, z1)) → c5(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(b, x1)) → c(F(mark(a), x1))
ACTIVE(f(b, x1)) → c
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVE(x1)) = [4]x1   
POL(F(x1, x2)) = 0   
POL(PROPER(x1)) = 0   
POL(TOP(x1)) = [4]x1   
POL(a) = 0   
POL(active(x1)) = 0   
POL(b) = [4]   
POL(c) = 0   
POL(c(x1)) = x1   
POL(c2(x1, x2)) = x1 + x2   
POL(c3(x1)) = x1   
POL(c4(x1)) = x1   
POL(c5(x1, x2, x3)) = x1 + x2 + x3   
POL(c8(x1, x2)) = x1 + x2   
POL(c9(x1, x2)) = x1 + x2   
POL(f(x1, x2)) = [2]x1   
POL(mark(x1)) = x1   
POL(ok(x1)) = x1   
POL(proper(x1)) = x1   

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, z0)) → mark(f(a, b))
active(b) → mark(a)
active(f(z0, z1)) → f(active(z0), z1)
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(a) → ok(a)
proper(b) → ok(b)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
PROPER(f(z0, z1)) → c5(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(b, x1)) → c(F(mark(a), x1))
ACTIVE(f(b, x1)) → c
S tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
PROPER(f(z0, z1)) → c5(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(b, x1)) → c
K tuples:

ACTIVE(f(b, x1)) → c(F(mark(a), x1))
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, PROPER, TOP, ACTIVE

Compound Symbols:

c3, c4, c5, c8, c9, c2, c, c

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

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

ACTIVE(f(b, x1)) → c
We considered the (Usable) Rules:

active(f(z0, z0)) → mark(f(a, b))
active(b) → mark(a)
active(f(z0, z1)) → f(active(z0), z1)
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(a) → ok(a)
proper(b) → ok(b)
And the Tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
PROPER(f(z0, z1)) → c5(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(b, x1)) → c(F(mark(a), x1))
ACTIVE(f(b, x1)) → c
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVE(x1)) = x1   
POL(F(x1, x2)) = 0   
POL(PROPER(x1)) = 0   
POL(TOP(x1)) = x1   
POL(a) = 0   
POL(active(x1)) = 0   
POL(b) = [1]   
POL(c) = 0   
POL(c(x1)) = x1   
POL(c2(x1, x2)) = x1 + x2   
POL(c3(x1)) = x1   
POL(c4(x1)) = x1   
POL(c5(x1, x2, x3)) = x1 + x2 + x3   
POL(c8(x1, x2)) = x1 + x2   
POL(c9(x1, x2)) = x1 + x2   
POL(f(x1, x2)) = [2]x1   
POL(mark(x1)) = x1   
POL(ok(x1)) = x1   
POL(proper(x1)) = x1   

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, z0)) → mark(f(a, b))
active(b) → mark(a)
active(f(z0, z1)) → f(active(z0), z1)
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(a) → ok(a)
proper(b) → ok(b)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
PROPER(f(z0, z1)) → c5(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(b, x1)) → c(F(mark(a), x1))
ACTIVE(f(b, x1)) → c
S tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
PROPER(f(z0, z1)) → c5(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
K tuples:

ACTIVE(f(b, x1)) → c(F(mark(a), x1))
ACTIVE(f(b, x1)) → c
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, PROPER, TOP, ACTIVE

Compound Symbols:

c3, c4, c5, c8, c9, c2, c, c

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

Use narrowing to replace PROPER(f(z0, z1)) → c5(F(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0), PROPER(a))
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0), PROPER(b))
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(a), PROPER(x1))
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(b), PROPER(x1))

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, z0)) → mark(f(a, b))
active(b) → mark(a)
active(f(z0, z1)) → f(active(z0), z1)
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(a) → ok(a)
proper(b) → ok(b)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(b, x1)) → c(F(mark(a), x1))
ACTIVE(f(b, x1)) → c
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0), PROPER(a))
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0), PROPER(b))
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(a), PROPER(x1))
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(b), PROPER(x1))
S tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0), PROPER(a))
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0), PROPER(b))
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(a), PROPER(x1))
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(b), PROPER(x1))
K tuples:

ACTIVE(f(b, x1)) → c(F(mark(a), x1))
ACTIVE(f(b, x1)) → c
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, TOP, ACTIVE, PROPER

Compound Symbols:

c3, c4, c8, c9, c2, c, c, c5

(17) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 1 of 14 dangling nodes:

ACTIVE(f(b, x1)) → c

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, z0)) → mark(f(a, b))
active(b) → mark(a)
active(f(z0, z1)) → f(active(z0), z1)
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(a) → ok(a)
proper(b) → ok(b)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(b, x1)) → c(F(mark(a), x1))
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0), PROPER(a))
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0), PROPER(b))
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(a), PROPER(x1))
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(b), PROPER(x1))
S tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0), PROPER(a))
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0), PROPER(b))
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(a), PROPER(x1))
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(b), PROPER(x1))
K tuples:

ACTIVE(f(b, x1)) → c(F(mark(a), x1))
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, TOP, ACTIVE, PROPER

Compound Symbols:

c3, c4, c8, c9, c2, c, c5

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

Removed 4 trailing tuple parts

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, z0)) → mark(f(a, b))
active(b) → mark(a)
active(f(z0, z1)) → f(active(z0), z1)
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(a) → ok(a)
proper(b) → ok(b)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(b, x1)) → c(F(mark(a), x1))
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(x1))
S tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(x1))
K tuples:

ACTIVE(f(b, x1)) → c(F(mark(a), x1))
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, TOP, ACTIVE, PROPER

Compound Symbols:

c3, c4, c8, c9, c2, c, c5, c5

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

Use narrowing to replace TOP(mark(z0)) → c8(TOP(proper(z0)), PROPER(z0)) by

TOP(mark(f(z0, z1))) → c8(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(mark(a)) → c8(TOP(ok(a)), PROPER(a))
TOP(mark(b)) → c8(TOP(ok(b)), PROPER(b))

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, z0)) → mark(f(a, b))
active(b) → mark(a)
active(f(z0, z1)) → f(active(z0), z1)
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(a) → ok(a)
proper(b) → ok(b)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(b, x1)) → c(F(mark(a), x1))
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(x1))
TOP(mark(f(z0, z1))) → c8(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(mark(a)) → c8(TOP(ok(a)), PROPER(a))
TOP(mark(b)) → c8(TOP(ok(b)), PROPER(b))
S tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(x1))
TOP(mark(f(z0, z1))) → c8(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(mark(a)) → c8(TOP(ok(a)), PROPER(a))
TOP(mark(b)) → c8(TOP(ok(b)), PROPER(b))
K tuples:

ACTIVE(f(b, x1)) → c(F(mark(a), x1))
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, TOP, ACTIVE, PROPER

Compound Symbols:

c3, c4, c9, c2, c, c5, c5, c8

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

Removed 2 trailing tuple parts

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, z0)) → mark(f(a, b))
active(b) → mark(a)
active(f(z0, z1)) → f(active(z0), z1)
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(a) → ok(a)
proper(b) → ok(b)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(b, x1)) → c(F(mark(a), x1))
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(x1))
TOP(mark(f(z0, z1))) → c8(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(mark(a)) → c8(TOP(ok(a)))
TOP(mark(b)) → c8(TOP(ok(b)))
S tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(x1))
TOP(mark(f(z0, z1))) → c8(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(mark(a)) → c8(TOP(ok(a)))
TOP(mark(b)) → c8(TOP(ok(b)))
K tuples:

ACTIVE(f(b, x1)) → c(F(mark(a), x1))
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, TOP, ACTIVE, PROPER

Compound Symbols:

c3, c4, c9, c2, c, c5, c5, c8, c8

(25) 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.

TOP(mark(b)) → c8(TOP(ok(b)))
We considered the (Usable) Rules:

proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(a) → ok(a)
proper(b) → ok(b)
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
active(f(z0, z0)) → mark(f(a, b))
active(b) → mark(a)
active(f(z0, z1)) → f(active(z0), z1)
And the Tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(b, x1)) → c(F(mark(a), x1))
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(x1))
TOP(mark(f(z0, z1))) → c8(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(mark(a)) → c8(TOP(ok(a)))
TOP(mark(b)) → c8(TOP(ok(b)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVE(x1)) = 0   
POL(F(x1, x2)) = 0   
POL(PROPER(x1)) = 0   
POL(TOP(x1)) = x1   
POL(a) = 0   
POL(active(x1)) = 0   
POL(b) = [4]   
POL(c(x1)) = x1   
POL(c2(x1, x2)) = x1 + x2   
POL(c3(x1)) = x1   
POL(c4(x1)) = x1   
POL(c5(x1, x2)) = x1 + x2   
POL(c5(x1, x2, x3)) = x1 + x2 + x3   
POL(c8(x1)) = x1   
POL(c8(x1, x2)) = x1 + x2   
POL(c9(x1, x2)) = x1 + x2   
POL(f(x1, x2)) = [4]x1   
POL(mark(x1)) = x1   
POL(ok(x1)) = 0   
POL(proper(x1)) = 0   

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, z0)) → mark(f(a, b))
active(b) → mark(a)
active(f(z0, z1)) → f(active(z0), z1)
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(a) → ok(a)
proper(b) → ok(b)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(b, x1)) → c(F(mark(a), x1))
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(x1))
TOP(mark(f(z0, z1))) → c8(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(mark(a)) → c8(TOP(ok(a)))
TOP(mark(b)) → c8(TOP(ok(b)))
S tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(x1))
TOP(mark(f(z0, z1))) → c8(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(mark(a)) → c8(TOP(ok(a)))
K tuples:

ACTIVE(f(b, x1)) → c(F(mark(a), x1))
TOP(mark(b)) → c8(TOP(ok(b)))
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, TOP, ACTIVE, PROPER

Compound Symbols:

c3, c4, c9, c2, c, c5, c5, c8, c8

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

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

TOP(mark(a)) → c8(TOP(ok(a)))
We considered the (Usable) Rules:

proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(a) → ok(a)
proper(b) → ok(b)
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
active(f(z0, z0)) → mark(f(a, b))
active(b) → mark(a)
active(f(z0, z1)) → f(active(z0), z1)
And the Tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(b, x1)) → c(F(mark(a), x1))
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(x1))
TOP(mark(f(z0, z1))) → c8(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(mark(a)) → c8(TOP(ok(a)))
TOP(mark(b)) → c8(TOP(ok(b)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVE(x1)) = 0   
POL(F(x1, x2)) = 0   
POL(PROPER(x1)) = 0   
POL(TOP(x1)) = [4]x1   
POL(a) = 0   
POL(active(x1)) = x1   
POL(b) = [1]   
POL(c(x1)) = x1   
POL(c2(x1, x2)) = x1 + x2   
POL(c3(x1)) = x1   
POL(c4(x1)) = x1   
POL(c5(x1, x2)) = x1 + x2   
POL(c5(x1, x2, x3)) = x1 + x2 + x3   
POL(c8(x1)) = x1   
POL(c8(x1, x2)) = x1 + x2   
POL(c9(x1, x2)) = x1 + x2   
POL(f(x1, x2)) = [1]   
POL(mark(x1)) = [1]   
POL(ok(x1)) = x1   
POL(proper(x1)) = 0   

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, z0)) → mark(f(a, b))
active(b) → mark(a)
active(f(z0, z1)) → f(active(z0), z1)
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(a) → ok(a)
proper(b) → ok(b)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(b, x1)) → c(F(mark(a), x1))
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(x1))
TOP(mark(f(z0, z1))) → c8(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(mark(a)) → c8(TOP(ok(a)))
TOP(mark(b)) → c8(TOP(ok(b)))
S tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0))
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(x1))
TOP(mark(f(z0, z1))) → c8(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
K tuples:

ACTIVE(f(b, x1)) → c(F(mark(a), x1))
TOP(mark(b)) → c8(TOP(ok(b)))
TOP(mark(a)) → c8(TOP(ok(a)))
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, TOP, ACTIVE, PROPER

Compound Symbols:

c3, c4, c9, c2, c, c5, c5, c8, c8

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

Use narrowing to replace TOP(ok(z0)) → c9(TOP(active(z0)), ACTIVE(z0)) by

TOP(ok(f(z0, z0))) → c9(TOP(mark(f(a, b))), ACTIVE(f(z0, z0)))
TOP(ok(b)) → c9(TOP(mark(a)), ACTIVE(b))
TOP(ok(f(z0, z1))) → c9(TOP(f(active(z0), z1)), ACTIVE(f(z0, z1)))

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, z0)) → mark(f(a, b))
active(b) → mark(a)
active(f(z0, z1)) → f(active(z0), z1)
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(a) → ok(a)
proper(b) → ok(b)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(b, x1)) → c(F(mark(a), x1))
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(x1))
TOP(mark(f(z0, z1))) → c8(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(mark(a)) → c8(TOP(ok(a)))
TOP(mark(b)) → c8(TOP(ok(b)))
TOP(ok(f(z0, z0))) → c9(TOP(mark(f(a, b))), ACTIVE(f(z0, z0)))
TOP(ok(b)) → c9(TOP(mark(a)), ACTIVE(b))
TOP(ok(f(z0, z1))) → c9(TOP(f(active(z0), z1)), ACTIVE(f(z0, z1)))
S tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(x1))
TOP(mark(f(z0, z1))) → c8(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(ok(f(z0, z0))) → c9(TOP(mark(f(a, b))), ACTIVE(f(z0, z0)))
TOP(ok(b)) → c9(TOP(mark(a)), ACTIVE(b))
TOP(ok(f(z0, z1))) → c9(TOP(f(active(z0), z1)), ACTIVE(f(z0, z1)))
K tuples:

ACTIVE(f(b, x1)) → c(F(mark(a), x1))
TOP(mark(b)) → c8(TOP(ok(b)))
TOP(mark(a)) → c8(TOP(ok(a)))
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, ACTIVE, PROPER, TOP

Compound Symbols:

c3, c4, c2, c, c5, c5, c8, c8, c9

(31) CdtUnreachableProof (EQUIVALENT transformation)

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

ACTIVE(f(f(z0, z0), x1)) → c2(F(mark(f(a, b)), x1), ACTIVE(f(z0, z0)))
ACTIVE(f(f(z0, z1), x1)) → c2(F(f(active(z0), z1), x1), ACTIVE(f(z0, z1)))
ACTIVE(f(b, x1)) → c(F(mark(a), x1))
PROPER(f(x0, f(z0, z1))) → c5(F(proper(x0), f(proper(z0), proper(z1))), PROPER(x0), PROPER(f(z0, z1)))
PROPER(f(f(z0, z1), x1)) → c5(F(f(proper(z0), proper(z1)), proper(x1)), PROPER(f(z0, z1)), PROPER(x1))
PROPER(f(x0, a)) → c5(F(proper(x0), ok(a)), PROPER(x0))
PROPER(f(x0, b)) → c5(F(proper(x0), ok(b)), PROPER(x0))
PROPER(f(a, x1)) → c5(F(ok(a), proper(x1)), PROPER(x1))
PROPER(f(b, x1)) → c5(F(ok(b), proper(x1)), PROPER(x1))
TOP(mark(f(z0, z1))) → c8(TOP(f(proper(z0), proper(z1))), PROPER(f(z0, z1)))
TOP(ok(f(z0, z0))) → c9(TOP(mark(f(a, b))), ACTIVE(f(z0, z0)))
TOP(ok(f(z0, z1))) → c9(TOP(f(active(z0), z1)), ACTIVE(f(z0, z1)))

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, z0)) → mark(f(a, b))
active(b) → mark(a)
active(f(z0, z1)) → f(active(z0), z1)
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(a) → ok(a)
proper(b) → ok(b)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
TOP(mark(a)) → c8(TOP(ok(a)))
TOP(mark(b)) → c8(TOP(ok(b)))
TOP(ok(b)) → c9(TOP(mark(a)), ACTIVE(b))
S tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
TOP(ok(b)) → c9(TOP(mark(a)), ACTIVE(b))
K tuples:

TOP(mark(b)) → c8(TOP(ok(b)))
TOP(mark(a)) → c8(TOP(ok(a)))
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F, TOP

Compound Symbols:

c3, c4, c8, c9

(33) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 3 of 5 dangling nodes:

TOP(ok(b)) → c9(TOP(mark(a)), ACTIVE(b))
TOP(mark(a)) → c8(TOP(ok(a)))
TOP(mark(b)) → c8(TOP(ok(b)))

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, z0)) → mark(f(a, b))
active(b) → mark(a)
active(f(z0, z1)) → f(active(z0), z1)
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(a) → ok(a)
proper(b) → ok(b)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
S tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
K tuples:none
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F

Compound Symbols:

c3, c4

(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.

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(F(x1, x2)) = [4]x1 + [4]x2   
POL(c3(x1)) = x1   
POL(c4(x1)) = x1   
POL(mark(x1)) = [2] + x1   
POL(ok(x1)) = [4] + x1   

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(f(z0, z0)) → mark(f(a, b))
active(b) → mark(a)
active(f(z0, z1)) → f(active(z0), z1)
f(mark(z0), z1) → mark(f(z0, z1))
f(ok(z0), ok(z1)) → ok(f(z0, z1))
proper(f(z0, z1)) → f(proper(z0), proper(z1))
proper(a) → ok(a)
proper(b) → ok(b)
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))
Tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
S tuples:none
K tuples:

F(mark(z0), z1) → c3(F(z0, z1))
F(ok(z0), ok(z1)) → c4(F(z0, z1))
Defined Rule Symbols:

active, f, proper, top

Defined Pair Symbols:

F

Compound Symbols:

c3, c4

(37) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(38) BOUNDS(O(1), O(1))