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:

is_empty(nil) → true
is_empty(cons(x, l)) → false
hd(cons(x, l)) → x
tl(cons(x, l)) → l
append(l1, l2) → ifappend(l1, l2, l1)
ifappend(l1, l2, nil) → l2
ifappend(l1, l2, cons(x, l)) → cons(x, append(l, l2))

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

is_empty(nil) → true
is_empty(cons(z0, z1)) → false
hd(cons(z0, z1)) → z0
tl(cons(z0, z1)) → z1
append(z0, z1) → ifappend(z0, z1, z0)
ifappend(z0, z1, nil) → z1
ifappend(z0, z1, cons(z2, z3)) → cons(z2, append(z3, z1))

Tuples:

APPEND(z0, z1) → c4(IFAPPEND(z0, z1, z0))
IFAPPEND(z0, z1, cons(z2, z3)) → c6(APPEND(z3, z1))

S tuples:

APPEND(z0, z1) → c4(IFAPPEND(z0, z1, z0))
IFAPPEND(z0, z1, cons(z2, z3)) → c6(APPEND(z3, z1))

K tuples:none
Defined Rule Symbols:

is_empty, hd, tl, append, ifappend

Defined Pair Symbols:

APPEND, IFAPPEND

Compound Symbols:

c4, c6

(3) 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.

IFAPPEND(z0, z1, cons(z2, z3)) → c6(APPEND(z3, z1))

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

APPEND(z0, z1) → c4(IFAPPEND(z0, z1, z0))
IFAPPEND(z0, z1, cons(z2, z3)) → c6(APPEND(z3, z1))

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

POL(APPEND(x₁, x₂)) = x₁   
POL(IFAPPEND(x₁, x₂, x₃)) = x₃   
POL(c4(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(cons(x₁, x₂)) = [1] + x₂   

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

is_empty(nil) → true
is_empty(cons(z0, z1)) → false
hd(cons(z0, z1)) → z0
tl(cons(z0, z1)) → z1
append(z0, z1) → ifappend(z0, z1, z0)
ifappend(z0, z1, nil) → z1
ifappend(z0, z1, cons(z2, z3)) → cons(z2, append(z3, z1))

Tuples:

APPEND(z0, z1) → c4(IFAPPEND(z0, z1, z0))
IFAPPEND(z0, z1, cons(z2, z3)) → c6(APPEND(z3, z1))

S tuples:

APPEND(z0, z1) → c4(IFAPPEND(z0, z1, z0))

K tuples:

IFAPPEND(z0, z1, cons(z2, z3)) → c6(APPEND(z3, z1))

Defined Rule Symbols:

is_empty, hd, tl, append, ifappend

Defined Pair Symbols:

APPEND, IFAPPEND

Compound Symbols:

c4, c6

(5) CdtKnowledgeProof (EQUIVALENT transformation)

The following tuples could be moved from S to K by knowledge propagation:

APPEND(z0, z1) → c4(IFAPPEND(z0, z1, z0))
IFAPPEND(z0, z1, cons(z2, z3)) → c6(APPEND(z3, z1))

Now S is empty

(6) BOUNDS(O(1), O(1))