### (0) Obligation:

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

a__f(0) → cons(0, f(s(0)))
a__f(s(0)) → a__f(a__p(s(0)))
a__p(s(0)) → 0
mark(f(X)) → a__f(mark(X))
mark(p(X)) → a__p(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__p(X) → p(X)

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

### (2) Obligation:

Complexity Dependency Tuples Problem
Rules:

a__f(0) → cons(0, f(s(0)))
a__f(s(0)) → a__f(a__p(s(0)))
a__f(z0) → f(z0)
a__p(s(0)) → 0
a__p(z0) → p(z0)
mark(f(z0)) → a__f(mark(z0))
mark(p(z0)) → a__p(mark(z0))
mark(0) → 0
mark(cons(z0, z1)) → cons(mark(z0), z1)
mark(s(z0)) → s(mark(z0))
Tuples:

A__F(0) → c
A__F(s(0)) → c1(A__F(a__p(s(0))), A__P(s(0)))
A__F(z0) → c2
A__P(s(0)) → c3
A__P(z0) → c4
MARK(f(z0)) → c5(A__F(mark(z0)), MARK(z0))
MARK(p(z0)) → c6(A__P(mark(z0)), MARK(z0))
MARK(0) → c7
MARK(cons(z0, z1)) → c8(MARK(z0))
MARK(s(z0)) → c9(MARK(z0))
S tuples:

A__F(0) → c
A__F(s(0)) → c1(A__F(a__p(s(0))), A__P(s(0)))
A__F(z0) → c2
A__P(s(0)) → c3
A__P(z0) → c4
MARK(f(z0)) → c5(A__F(mark(z0)), MARK(z0))
MARK(p(z0)) → c6(A__P(mark(z0)), MARK(z0))
MARK(0) → c7
MARK(cons(z0, z1)) → c8(MARK(z0))
MARK(s(z0)) → c9(MARK(z0))
K tuples:none
Defined Rule Symbols:

a__f, a__p, mark

Defined Pair Symbols:

A__F, A__P, MARK

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9

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

Removed 5 trailing nodes:

A__F(0) → c
A__F(z0) → c2
A__P(s(0)) → c3
A__P(z0) → c4
MARK(0) → c7

### (4) Obligation:

Complexity Dependency Tuples Problem
Rules:

a__f(0) → cons(0, f(s(0)))
a__f(s(0)) → a__f(a__p(s(0)))
a__f(z0) → f(z0)
a__p(s(0)) → 0
a__p(z0) → p(z0)
mark(f(z0)) → a__f(mark(z0))
mark(p(z0)) → a__p(mark(z0))
mark(0) → 0
mark(cons(z0, z1)) → cons(mark(z0), z1)
mark(s(z0)) → s(mark(z0))
Tuples:

A__F(s(0)) → c1(A__F(a__p(s(0))), A__P(s(0)))
MARK(f(z0)) → c5(A__F(mark(z0)), MARK(z0))
MARK(p(z0)) → c6(A__P(mark(z0)), MARK(z0))
MARK(cons(z0, z1)) → c8(MARK(z0))
MARK(s(z0)) → c9(MARK(z0))
S tuples:

A__F(s(0)) → c1(A__F(a__p(s(0))), A__P(s(0)))
MARK(f(z0)) → c5(A__F(mark(z0)), MARK(z0))
MARK(p(z0)) → c6(A__P(mark(z0)), MARK(z0))
MARK(cons(z0, z1)) → c8(MARK(z0))
MARK(s(z0)) → c9(MARK(z0))
K tuples:none
Defined Rule Symbols:

a__f, a__p, mark

Defined Pair Symbols:

A__F, MARK

Compound Symbols:

c1, c5, c6, c8, c9

### (5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing tuple parts

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:

a__f(0) → cons(0, f(s(0)))
a__f(s(0)) → a__f(a__p(s(0)))
a__f(z0) → f(z0)
a__p(s(0)) → 0
a__p(z0) → p(z0)
mark(f(z0)) → a__f(mark(z0))
mark(p(z0)) → a__p(mark(z0))
mark(0) → 0
mark(cons(z0, z1)) → cons(mark(z0), z1)
mark(s(z0)) → s(mark(z0))
Tuples:

MARK(f(z0)) → c5(A__F(mark(z0)), MARK(z0))
MARK(cons(z0, z1)) → c8(MARK(z0))
MARK(s(z0)) → c9(MARK(z0))
A__F(s(0)) → c1(A__F(a__p(s(0))))
MARK(p(z0)) → c6(MARK(z0))
S tuples:

MARK(f(z0)) → c5(A__F(mark(z0)), MARK(z0))
MARK(cons(z0, z1)) → c8(MARK(z0))
MARK(s(z0)) → c9(MARK(z0))
A__F(s(0)) → c1(A__F(a__p(s(0))))
MARK(p(z0)) → c6(MARK(z0))
K tuples:none
Defined Rule Symbols:

a__f, a__p, mark

Defined Pair Symbols:

MARK, A__F

Compound Symbols:

c5, c8, c9, c1, c6

### (7) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)

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

MARK(f(z0)) → c5(A__F(mark(z0)), MARK(z0))
MARK(cons(z0, z1)) → c8(MARK(z0))
MARK(p(z0)) → c6(MARK(z0))
We considered the (Usable) Rules:none
And the Tuples:

MARK(f(z0)) → c5(A__F(mark(z0)), MARK(z0))
MARK(cons(z0, z1)) → c8(MARK(z0))
MARK(s(z0)) → c9(MARK(z0))
A__F(s(0)) → c1(A__F(a__p(s(0))))
MARK(p(z0)) → c6(MARK(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(A__F(x1)) = 0
POL(MARK(x1)) = [4]x1
POL(a__f(x1)) = [3] + [2]x1
POL(a__p(x1)) = [2]
POL(c1(x1)) = x1
POL(c5(x1, x2)) = x1 + x2
POL(c6(x1)) = x1
POL(c8(x1)) = x1
POL(c9(x1)) = x1
POL(cons(x1, x2)) = [4] + x1
POL(f(x1)) = [5] + x1
POL(mark(x1)) = [2] + [2]x1
POL(p(x1)) = [5] + x1
POL(s(x1)) = x1

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:

a__f(0) → cons(0, f(s(0)))
a__f(s(0)) → a__f(a__p(s(0)))
a__f(z0) → f(z0)
a__p(s(0)) → 0
a__p(z0) → p(z0)
mark(f(z0)) → a__f(mark(z0))
mark(p(z0)) → a__p(mark(z0))
mark(0) → 0
mark(cons(z0, z1)) → cons(mark(z0), z1)
mark(s(z0)) → s(mark(z0))
Tuples:

MARK(f(z0)) → c5(A__F(mark(z0)), MARK(z0))
MARK(cons(z0, z1)) → c8(MARK(z0))
MARK(s(z0)) → c9(MARK(z0))
A__F(s(0)) → c1(A__F(a__p(s(0))))
MARK(p(z0)) → c6(MARK(z0))
S tuples:

MARK(s(z0)) → c9(MARK(z0))
A__F(s(0)) → c1(A__F(a__p(s(0))))
K tuples:

MARK(f(z0)) → c5(A__F(mark(z0)), MARK(z0))
MARK(cons(z0, z1)) → c8(MARK(z0))
MARK(p(z0)) → c6(MARK(z0))
Defined Rule Symbols:

a__f, a__p, mark

Defined Pair Symbols:

MARK, A__F

Compound Symbols:

c5, c8, c9, c1, c6

### (9) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)

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

MARK(s(z0)) → c9(MARK(z0))
We considered the (Usable) Rules:none
And the Tuples:

MARK(f(z0)) → c5(A__F(mark(z0)), MARK(z0))
MARK(cons(z0, z1)) → c8(MARK(z0))
MARK(s(z0)) → c9(MARK(z0))
A__F(s(0)) → c1(A__F(a__p(s(0))))
MARK(p(z0)) → c6(MARK(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(A__F(x1)) = 0
POL(MARK(x1)) = [3]x1
POL(a__f(x1)) = [3] + [2]x1
POL(a__p(x1)) = 0
POL(c1(x1)) = x1
POL(c5(x1, x2)) = x1 + x2
POL(c6(x1)) = x1
POL(c8(x1)) = x1
POL(c9(x1)) = x1
POL(cons(x1, x2)) = [5] + x1
POL(f(x1)) = x1
POL(mark(x1)) = [2] + [3]x1
POL(p(x1)) = [2] + x1
POL(s(x1)) = [3] + x1

### (10) Obligation:

Complexity Dependency Tuples Problem
Rules:

a__f(0) → cons(0, f(s(0)))
a__f(s(0)) → a__f(a__p(s(0)))
a__f(z0) → f(z0)
a__p(s(0)) → 0
a__p(z0) → p(z0)
mark(f(z0)) → a__f(mark(z0))
mark(p(z0)) → a__p(mark(z0))
mark(0) → 0
mark(cons(z0, z1)) → cons(mark(z0), z1)
mark(s(z0)) → s(mark(z0))
Tuples:

MARK(f(z0)) → c5(A__F(mark(z0)), MARK(z0))
MARK(cons(z0, z1)) → c8(MARK(z0))
MARK(s(z0)) → c9(MARK(z0))
A__F(s(0)) → c1(A__F(a__p(s(0))))
MARK(p(z0)) → c6(MARK(z0))
S tuples:

A__F(s(0)) → c1(A__F(a__p(s(0))))
K tuples:

MARK(f(z0)) → c5(A__F(mark(z0)), MARK(z0))
MARK(cons(z0, z1)) → c8(MARK(z0))
MARK(p(z0)) → c6(MARK(z0))
MARK(s(z0)) → c9(MARK(z0))
Defined Rule Symbols:

a__f, a__p, mark

Defined Pair Symbols:

MARK, A__F

Compound Symbols:

c5, c8, c9, c1, c6

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

Use narrowing to replace MARK(f(z0)) → c5(A__F(mark(z0)), MARK(z0)) by

MARK(f(f(z0))) → c5(A__F(a__f(mark(z0))), MARK(f(z0)))
MARK(f(p(z0))) → c5(A__F(a__p(mark(z0))), MARK(p(z0)))
MARK(f(0)) → c5(A__F(0), MARK(0))
MARK(f(cons(z0, z1))) → c5(A__F(cons(mark(z0), z1)), MARK(cons(z0, z1)))
MARK(f(s(z0))) → c5(A__F(s(mark(z0))), MARK(s(z0)))

### (12) Obligation:

Complexity Dependency Tuples Problem
Rules:

a__f(0) → cons(0, f(s(0)))
a__f(s(0)) → a__f(a__p(s(0)))
a__f(z0) → f(z0)
a__p(s(0)) → 0
a__p(z0) → p(z0)
mark(f(z0)) → a__f(mark(z0))
mark(p(z0)) → a__p(mark(z0))
mark(0) → 0
mark(cons(z0, z1)) → cons(mark(z0), z1)
mark(s(z0)) → s(mark(z0))
Tuples:

MARK(cons(z0, z1)) → c8(MARK(z0))
MARK(s(z0)) → c9(MARK(z0))
A__F(s(0)) → c1(A__F(a__p(s(0))))
MARK(p(z0)) → c6(MARK(z0))
MARK(f(f(z0))) → c5(A__F(a__f(mark(z0))), MARK(f(z0)))
MARK(f(p(z0))) → c5(A__F(a__p(mark(z0))), MARK(p(z0)))
MARK(f(0)) → c5(A__F(0), MARK(0))
MARK(f(cons(z0, z1))) → c5(A__F(cons(mark(z0), z1)), MARK(cons(z0, z1)))
MARK(f(s(z0))) → c5(A__F(s(mark(z0))), MARK(s(z0)))
S tuples:

A__F(s(0)) → c1(A__F(a__p(s(0))))
K tuples:

MARK(f(z0)) → c5(A__F(mark(z0)), MARK(z0))
MARK(cons(z0, z1)) → c8(MARK(z0))
MARK(p(z0)) → c6(MARK(z0))
MARK(s(z0)) → c9(MARK(z0))
Defined Rule Symbols:

a__f, a__p, mark

Defined Pair Symbols:

MARK, A__F

Compound Symbols:

c8, c9, c1, c6, c5

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

Removed 1 trailing nodes:

MARK(f(0)) → c5(A__F(0), MARK(0))

### (14) Obligation:

Complexity Dependency Tuples Problem
Rules:

a__f(0) → cons(0, f(s(0)))
a__f(s(0)) → a__f(a__p(s(0)))
a__f(z0) → f(z0)
a__p(s(0)) → 0
a__p(z0) → p(z0)
mark(f(z0)) → a__f(mark(z0))
mark(p(z0)) → a__p(mark(z0))
mark(0) → 0
mark(cons(z0, z1)) → cons(mark(z0), z1)
mark(s(z0)) → s(mark(z0))
Tuples:

MARK(cons(z0, z1)) → c8(MARK(z0))
MARK(s(z0)) → c9(MARK(z0))
A__F(s(0)) → c1(A__F(a__p(s(0))))
MARK(p(z0)) → c6(MARK(z0))
MARK(f(f(z0))) → c5(A__F(a__f(mark(z0))), MARK(f(z0)))
MARK(f(p(z0))) → c5(A__F(a__p(mark(z0))), MARK(p(z0)))
MARK(f(cons(z0, z1))) → c5(A__F(cons(mark(z0), z1)), MARK(cons(z0, z1)))
MARK(f(s(z0))) → c5(A__F(s(mark(z0))), MARK(s(z0)))
S tuples:

A__F(s(0)) → c1(A__F(a__p(s(0))))
K tuples:

MARK(cons(z0, z1)) → c8(MARK(z0))
MARK(p(z0)) → c6(MARK(z0))
MARK(s(z0)) → c9(MARK(z0))
Defined Rule Symbols:

a__f, a__p, mark

Defined Pair Symbols:

MARK, A__F

Compound Symbols:

c8, c9, c1, c6, c5

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

Removed 1 trailing tuple parts

### (16) Obligation:

Complexity Dependency Tuples Problem
Rules:

a__f(0) → cons(0, f(s(0)))
a__f(s(0)) → a__f(a__p(s(0)))
a__f(z0) → f(z0)
a__p(s(0)) → 0
a__p(z0) → p(z0)
mark(f(z0)) → a__f(mark(z0))
mark(p(z0)) → a__p(mark(z0))
mark(0) → 0
mark(cons(z0, z1)) → cons(mark(z0), z1)
mark(s(z0)) → s(mark(z0))
Tuples:

MARK(cons(z0, z1)) → c8(MARK(z0))
MARK(s(z0)) → c9(MARK(z0))
A__F(s(0)) → c1(A__F(a__p(s(0))))
MARK(p(z0)) → c6(MARK(z0))
MARK(f(f(z0))) → c5(A__F(a__f(mark(z0))), MARK(f(z0)))
MARK(f(p(z0))) → c5(A__F(a__p(mark(z0))), MARK(p(z0)))
MARK(f(s(z0))) → c5(A__F(s(mark(z0))), MARK(s(z0)))
MARK(f(cons(z0, z1))) → c5(MARK(cons(z0, z1)))
S tuples:

A__F(s(0)) → c1(A__F(a__p(s(0))))
K tuples:

MARK(cons(z0, z1)) → c8(MARK(z0))
MARK(p(z0)) → c6(MARK(z0))
MARK(s(z0)) → c9(MARK(z0))
Defined Rule Symbols:

a__f, a__p, mark

Defined Pair Symbols:

MARK, A__F

Compound Symbols:

c8, c9, c1, c6, c5, c5

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

Use narrowing to replace A__F(s(0)) → c1(A__F(a__p(s(0)))) by

A__F(s(0)) → c1(A__F(0))
A__F(s(0)) → c1(A__F(p(s(0))))

### (18) Obligation:

Complexity Dependency Tuples Problem
Rules:

a__f(0) → cons(0, f(s(0)))
a__f(s(0)) → a__f(a__p(s(0)))
a__f(z0) → f(z0)
a__p(s(0)) → 0
a__p(z0) → p(z0)
mark(f(z0)) → a__f(mark(z0))
mark(p(z0)) → a__p(mark(z0))
mark(0) → 0
mark(cons(z0, z1)) → cons(mark(z0), z1)
mark(s(z0)) → s(mark(z0))
Tuples:

MARK(cons(z0, z1)) → c8(MARK(z0))
MARK(s(z0)) → c9(MARK(z0))
MARK(p(z0)) → c6(MARK(z0))
MARK(f(f(z0))) → c5(A__F(a__f(mark(z0))), MARK(f(z0)))
MARK(f(p(z0))) → c5(A__F(a__p(mark(z0))), MARK(p(z0)))
MARK(f(s(z0))) → c5(A__F(s(mark(z0))), MARK(s(z0)))
MARK(f(cons(z0, z1))) → c5(MARK(cons(z0, z1)))
A__F(s(0)) → c1(A__F(0))
A__F(s(0)) → c1(A__F(p(s(0))))
S tuples:

A__F(s(0)) → c1(A__F(0))
A__F(s(0)) → c1(A__F(p(s(0))))
K tuples:

MARK(cons(z0, z1)) → c8(MARK(z0))
MARK(p(z0)) → c6(MARK(z0))
MARK(s(z0)) → c9(MARK(z0))
Defined Rule Symbols:

a__f, a__p, mark

Defined Pair Symbols:

MARK, A__F

Compound Symbols:

c8, c9, c6, c5, c5, c1

### (19) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing tuple parts

### (20) Obligation:

Complexity Dependency Tuples Problem
Rules:

a__f(0) → cons(0, f(s(0)))
a__f(s(0)) → a__f(a__p(s(0)))
a__f(z0) → f(z0)
a__p(s(0)) → 0
a__p(z0) → p(z0)
mark(f(z0)) → a__f(mark(z0))
mark(p(z0)) → a__p(mark(z0))
mark(0) → 0
mark(cons(z0, z1)) → cons(mark(z0), z1)
mark(s(z0)) → s(mark(z0))
Tuples:

MARK(cons(z0, z1)) → c8(MARK(z0))
MARK(s(z0)) → c9(MARK(z0))
MARK(p(z0)) → c6(MARK(z0))
MARK(f(f(z0))) → c5(A__F(a__f(mark(z0))), MARK(f(z0)))
MARK(f(p(z0))) → c5(A__F(a__p(mark(z0))), MARK(p(z0)))
MARK(f(s(z0))) → c5(A__F(s(mark(z0))), MARK(s(z0)))
MARK(f(cons(z0, z1))) → c5(MARK(cons(z0, z1)))
A__F(s(0)) → c1
S tuples:

A__F(s(0)) → c1
K tuples:

MARK(cons(z0, z1)) → c8(MARK(z0))
MARK(p(z0)) → c6(MARK(z0))
MARK(s(z0)) → c9(MARK(z0))
Defined Rule Symbols:

a__f, a__p, mark

Defined Pair Symbols:

MARK, A__F

Compound Symbols:

c8, c9, c6, c5, c5, c1

### (21) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)

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

A__F(s(0)) → c1
We considered the (Usable) Rules:none
And the Tuples:

MARK(cons(z0, z1)) → c8(MARK(z0))
MARK(s(z0)) → c9(MARK(z0))
MARK(p(z0)) → c6(MARK(z0))
MARK(f(f(z0))) → c5(A__F(a__f(mark(z0))), MARK(f(z0)))
MARK(f(p(z0))) → c5(A__F(a__p(mark(z0))), MARK(p(z0)))
MARK(f(s(z0))) → c5(A__F(s(mark(z0))), MARK(s(z0)))
MARK(f(cons(z0, z1))) → c5(MARK(cons(z0, z1)))
A__F(s(0)) → c1
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(A__F(x1)) = [1]
POL(MARK(x1)) = x1
POL(a__f(x1)) = 0
POL(a__p(x1)) = 0
POL(c1) = 0
POL(c5(x1)) = x1
POL(c5(x1, x2)) = x1 + x2
POL(c6(x1)) = x1
POL(c8(x1)) = x1
POL(c9(x1)) = x1
POL(cons(x1, x2)) = x1
POL(f(x1)) = [1] + x1
POL(mark(x1)) = 0
POL(p(x1)) = x1
POL(s(x1)) = x1

### (22) Obligation:

Complexity Dependency Tuples Problem
Rules:

a__f(0) → cons(0, f(s(0)))
a__f(s(0)) → a__f(a__p(s(0)))
a__f(z0) → f(z0)
a__p(s(0)) → 0
a__p(z0) → p(z0)
mark(f(z0)) → a__f(mark(z0))
mark(p(z0)) → a__p(mark(z0))
mark(0) → 0
mark(cons(z0, z1)) → cons(mark(z0), z1)
mark(s(z0)) → s(mark(z0))
Tuples:

MARK(cons(z0, z1)) → c8(MARK(z0))
MARK(s(z0)) → c9(MARK(z0))
MARK(p(z0)) → c6(MARK(z0))
MARK(f(f(z0))) → c5(A__F(a__f(mark(z0))), MARK(f(z0)))
MARK(f(p(z0))) → c5(A__F(a__p(mark(z0))), MARK(p(z0)))
MARK(f(s(z0))) → c5(A__F(s(mark(z0))), MARK(s(z0)))
MARK(f(cons(z0, z1))) → c5(MARK(cons(z0, z1)))
A__F(s(0)) → c1
S tuples:none
K tuples:

MARK(cons(z0, z1)) → c8(MARK(z0))
MARK(p(z0)) → c6(MARK(z0))
MARK(s(z0)) → c9(MARK(z0))
A__F(s(0)) → c1
Defined Rule Symbols:

a__f, a__p, mark

Defined Pair Symbols:

MARK, A__F

Compound Symbols:

c8, c9, c6, c5, c5, c1

### (23) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

The set S is empty