### (0) Obligation:

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

decrease(Cons(x, xs)) → decrease(xs)
decrease(Nil) → number42(Nil)
number42(x) → Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))))))))))))))))))))))))))))
goal(x) → decrease(x)

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

### (2) Obligation:

Complexity Dependency Tuples Problem
Rules:

decrease(Cons(z0, z1)) → decrease(z1)
decrease(Nil) → number42(Nil)
number42(z0) → Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))))))))))))))))))))))))))))
goal(z0) → decrease(z0)
Tuples:

DECREASE(Cons(z0, z1)) → c(DECREASE(z1))
DECREASE(Nil) → c1(NUMBER42(Nil))
NUMBER42(z0) → c2
GOAL(z0) → c3(DECREASE(z0))
S tuples:

DECREASE(Cons(z0, z1)) → c(DECREASE(z1))
DECREASE(Nil) → c1(NUMBER42(Nil))
NUMBER42(z0) → c2
GOAL(z0) → c3(DECREASE(z0))
K tuples:none
Defined Rule Symbols:

decrease, number42, goal

Defined Pair Symbols:

DECREASE, NUMBER42, GOAL

Compound Symbols:

c, c1, c2, c3

### (3) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

GOAL(z0) → c3(DECREASE(z0))
Removed 2 trailing nodes:

NUMBER42(z0) → c2
DECREASE(Nil) → c1(NUMBER42(Nil))

### (4) Obligation:

Complexity Dependency Tuples Problem
Rules:

decrease(Cons(z0, z1)) → decrease(z1)
decrease(Nil) → number42(Nil)
number42(z0) → Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))))))))))))))))))))))))))))
goal(z0) → decrease(z0)
Tuples:

DECREASE(Cons(z0, z1)) → c(DECREASE(z1))
S tuples:

DECREASE(Cons(z0, z1)) → c(DECREASE(z1))
K tuples:none
Defined Rule Symbols:

decrease, number42, goal

Defined Pair Symbols:

DECREASE

Compound Symbols:

c

### (5) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

decrease(Cons(z0, z1)) → decrease(z1)
decrease(Nil) → number42(Nil)
number42(z0) → Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))))))))))))))))))))))))))))
goal(z0) → decrease(z0)

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

DECREASE(Cons(z0, z1)) → c(DECREASE(z1))
S tuples:

DECREASE(Cons(z0, z1)) → c(DECREASE(z1))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

DECREASE

Compound Symbols:

c

### (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.

DECREASE(Cons(z0, z1)) → c(DECREASE(z1))
We considered the (Usable) Rules:none
And the Tuples:

DECREASE(Cons(z0, z1)) → c(DECREASE(z1))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(Cons(x1, x2)) = [1] + x2
POL(DECREASE(x1)) = [5]x1
POL(c(x1)) = x1

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

DECREASE(Cons(z0, z1)) → c(DECREASE(z1))
S tuples:none
K tuples:

DECREASE(Cons(z0, z1)) → c(DECREASE(z1))
Defined Rule Symbols:none

Defined Pair Symbols:

DECREASE

Compound Symbols:

c

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

The set S is empty