### (0) Obligation:

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

list(Cons(x, xs)) → list(xs)
list(Nil) → True
list(Nil) → isEmpty[Match](Nil)
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(x) → list(x)

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

### (2) Obligation:

Complexity Dependency Tuples Problem
Rules:

list(Cons(z0, z1)) → list(z1)
list(Nil) → True
list(Nil) → isEmpty[Match](Nil)
notEmpty(Cons(z0, z1)) → True
notEmpty(Nil) → False
goal(z0) → list(z0)
Tuples:

LIST(Cons(z0, z1)) → c(LIST(z1))
LIST(Nil) → c1
LIST(Nil) → c2
NOTEMPTY(Cons(z0, z1)) → c3
NOTEMPTY(Nil) → c4
GOAL(z0) → c5(LIST(z0))
S tuples:

LIST(Cons(z0, z1)) → c(LIST(z1))
LIST(Nil) → c1
LIST(Nil) → c2
NOTEMPTY(Cons(z0, z1)) → c3
NOTEMPTY(Nil) → c4
GOAL(z0) → c5(LIST(z0))
K tuples:none
Defined Rule Symbols:

list, notEmpty, goal

Defined Pair Symbols:

LIST, NOTEMPTY, GOAL

Compound Symbols:

c, c1, c2, c3, c4, c5

### (3) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

GOAL(z0) → c5(LIST(z0))
Removed 4 trailing nodes:

NOTEMPTY(Nil) → c4
NOTEMPTY(Cons(z0, z1)) → c3
LIST(Nil) → c2
LIST(Nil) → c1

### (4) Obligation:

Complexity Dependency Tuples Problem
Rules:

list(Cons(z0, z1)) → list(z1)
list(Nil) → True
list(Nil) → isEmpty[Match](Nil)
notEmpty(Cons(z0, z1)) → True
notEmpty(Nil) → False
goal(z0) → list(z0)
Tuples:

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

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

list, notEmpty, goal

Defined Pair Symbols:

LIST

Compound Symbols:

c

### (5) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

list(Cons(z0, z1)) → list(z1)
list(Nil) → True
list(Nil) → isEmpty[Match](Nil)
notEmpty(Cons(z0, z1)) → True
notEmpty(Nil) → False
goal(z0) → list(z0)

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

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

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

Defined Pair Symbols:

LIST

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.

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

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

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

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

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

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

Defined Pair Symbols:

LIST

Compound Symbols:

c

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

The set S is empty