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

(0) Obligation:

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

addlist(Cons(x, xs'), Cons(S(0), xs)) → Cons(S(x), addlist(xs', xs))
addlist(Cons(S(0), xs'), Cons(x, xs)) → Cons(S(x), addlist(xs', xs))
addlist(Nil, ys) → Nil
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(xs, ys) → addlist(xs, ys)

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

addlist(Cons(z0, z1), Cons(S(0), z2)) → Cons(S(z0), addlist(z1, z2))
addlist(Cons(S(0), z0), Cons(z1, z2)) → Cons(S(z1), addlist(z0, z2))
addlist(Nil, z0) → Nil
notEmpty(Cons(z0, z1)) → True
notEmpty(Nil) → False
goal(z0, z1) → addlist(z0, z1)

Tuples:

ADDLIST(Cons(z0, z1), Cons(S(0), z2)) → c(ADDLIST(z1, z2))
ADDLIST(Cons(S(0), z0), Cons(z1, z2)) → c1(ADDLIST(z0, z2))
ADDLIST(Nil, z0) → c2
NOTEMPTY(Cons(z0, z1)) → c3
NOTEMPTY(Nil) → c4
GOAL(z0, z1) → c5(ADDLIST(z0, z1))

S tuples:

ADDLIST(Cons(z0, z1), Cons(S(0), z2)) → c(ADDLIST(z1, z2))
ADDLIST(Cons(S(0), z0), Cons(z1, z2)) → c1(ADDLIST(z0, z2))
ADDLIST(Nil, z0) → c2
NOTEMPTY(Cons(z0, z1)) → c3
NOTEMPTY(Nil) → c4
GOAL(z0, z1) → c5(ADDLIST(z0, z1))

K tuples:none
Defined Rule Symbols:

addlist, notEmpty, goal

Defined Pair Symbols:

ADDLIST, NOTEMPTY, GOAL

Compound Symbols:

c, c1, c2, c3, c4, c5

(3) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 1 leading nodes:

GOAL(z0, z1) → c5(ADDLIST(z0, z1))

Removed 3 trailing nodes:

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

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

addlist(Cons(z0, z1), Cons(S(0), z2)) → Cons(S(z0), addlist(z1, z2))
addlist(Cons(S(0), z0), Cons(z1, z2)) → Cons(S(z1), addlist(z0, z2))
addlist(Nil, z0) → Nil
notEmpty(Cons(z0, z1)) → True
notEmpty(Nil) → False
goal(z0, z1) → addlist(z0, z1)

Tuples:

ADDLIST(Cons(z0, z1), Cons(S(0), z2)) → c(ADDLIST(z1, z2))
ADDLIST(Cons(S(0), z0), Cons(z1, z2)) → c1(ADDLIST(z0, z2))

S tuples:

ADDLIST(Cons(z0, z1), Cons(S(0), z2)) → c(ADDLIST(z1, z2))
ADDLIST(Cons(S(0), z0), Cons(z1, z2)) → c1(ADDLIST(z0, z2))

K tuples:none
Defined Rule Symbols:

addlist, notEmpty, goal

Defined Pair Symbols:

ADDLIST

Compound Symbols:

c, c1

(5) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

addlist(Cons(z0, z1), Cons(S(0), z2)) → Cons(S(z0), addlist(z1, z2))
addlist(Cons(S(0), z0), Cons(z1, z2)) → Cons(S(z1), addlist(z0, z2))
addlist(Nil, z0) → Nil
notEmpty(Cons(z0, z1)) → True
notEmpty(Nil) → False
goal(z0, z1) → addlist(z0, z1)

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

ADDLIST(Cons(z0, z1), Cons(S(0), z2)) → c(ADDLIST(z1, z2))
ADDLIST(Cons(S(0), z0), Cons(z1, z2)) → c1(ADDLIST(z0, z2))

S tuples:

ADDLIST(Cons(z0, z1), Cons(S(0), z2)) → c(ADDLIST(z1, z2))
ADDLIST(Cons(S(0), z0), Cons(z1, z2)) → c1(ADDLIST(z0, z2))

K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

ADDLIST

Compound Symbols:

c, c1

(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.

ADDLIST(Cons(z0, z1), Cons(S(0), z2)) → c(ADDLIST(z1, z2))
ADDLIST(Cons(S(0), z0), Cons(z1, z2)) → c1(ADDLIST(z0, z2))

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

ADDLIST(Cons(z0, z1), Cons(S(0), z2)) → c(ADDLIST(z1, z2))
ADDLIST(Cons(S(0), z0), Cons(z1, z2)) → c1(ADDLIST(z0, z2))

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

POL(0) = [2]   
POL(ADDLIST(x₁, x₂)) = [2]x₂   
POL(Cons(x₁, x₂)) = [3] + x₂   
POL(S(x₁)) = [3]   
POL(c(x₁)) = x₁   
POL(c1(x₁)) = x₁   

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

ADDLIST(Cons(z0, z1), Cons(S(0), z2)) → c(ADDLIST(z1, z2))
ADDLIST(Cons(S(0), z0), Cons(z1, z2)) → c1(ADDLIST(z0, z2))

S tuples:none
K tuples:

ADDLIST(Cons(z0, z1), Cons(S(0), z2)) → c(ADDLIST(z1, z2))
ADDLIST(Cons(S(0), z0), Cons(z1, z2)) → c1(ADDLIST(z0, z2))

Defined Rule Symbols:none

Defined Pair Symbols:

ADDLIST

Compound Symbols:

c, c1

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

The set S is empty

(10) BOUNDS(O(1), O(1))