### (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 Cpx (relative) TRS 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:

IS_EMPTY(nil) → c
IS_EMPTY(cons(z0, z1)) → c1
HD(cons(z0, z1)) → c2
TL(cons(z0, z1)) → c3
APPEND(z0, z1) → c4(IFAPPEND(z0, z1, z0))
IFAPPEND(z0, z1, nil) → c5
IFAPPEND(z0, z1, cons(z2, z3)) → c6(APPEND(z3, z1))
S tuples:

IS_EMPTY(nil) → c
IS_EMPTY(cons(z0, z1)) → c1
HD(cons(z0, z1)) → c2
TL(cons(z0, z1)) → c3
APPEND(z0, z1) → c4(IFAPPEND(z0, z1, z0))
IFAPPEND(z0, z1, nil) → c5
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:

IS_EMPTY, HD, TL, APPEND, IFAPPEND

Compound Symbols:

c, c1, c2, c3, c4, c5, c6

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

Removed 5 trailing nodes:

IS_EMPTY(nil) → c
IS_EMPTY(cons(z0, z1)) → c1
IFAPPEND(z0, z1, nil) → c5
HD(cons(z0, z1)) → c2
TL(cons(z0, z1)) → c3

### (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))
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

### (5) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

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))

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:none
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:none

Defined Pair Symbols:

APPEND, IFAPPEND

Compound Symbols:

c4, 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.

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

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:none
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:none

Defined Pair Symbols:

APPEND, IFAPPEND

Compound Symbols:

c4, c6

### (9) 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