### (0) Obligation:

The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^3).

The TRS R consists of the following rules:

app(nil, y) → y
reverse(nil) → nil
shuffle(nil) → nil

Rewrite Strategy: FULL

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

Converted rc-obligation to irc-obligation.

As the TRS is a non-duplicating overlay system, we have rc = irc.

### (2) Obligation:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^3).

The TRS R consists of the following rules:

app(nil, y) → y
reverse(nil) → nil
shuffle(nil) → nil

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

### (4) Obligation:

Complexity Dependency Tuples Problem
Rules:

app(nil, z0) → z0
reverse(nil) → nil
shuffle(nil) → nil
Tuples:

APP(nil, z0) → c
APP(add(z0, z1), z2) → c1(APP(z1, z2))
REVERSE(nil) → c2
SHUFFLE(nil) → c4
S tuples:

APP(nil, z0) → c
APP(add(z0, z1), z2) → c1(APP(z1, z2))
REVERSE(nil) → c2
SHUFFLE(nil) → c4
K tuples:none
Defined Rule Symbols:

app, reverse, shuffle

Defined Pair Symbols:

APP, REVERSE, SHUFFLE

Compound Symbols:

c, c1, c2, c3, c4, c5

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

Removed 3 trailing nodes:

APP(nil, z0) → c
SHUFFLE(nil) → c4
REVERSE(nil) → c2

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:

app(nil, z0) → z0
reverse(nil) → nil
shuffle(nil) → nil
Tuples:

APP(add(z0, z1), z2) → c1(APP(z1, z2))
S tuples:

APP(add(z0, z1), z2) → c1(APP(z1, z2))
K tuples:none
Defined Rule Symbols:

app, reverse, shuffle

Defined Pair Symbols:

APP, REVERSE, SHUFFLE

Compound Symbols:

c1, c3, c5

### (7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

shuffle(nil) → nil

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:

reverse(nil) → nil
app(nil, z0) → z0
Tuples:

APP(add(z0, z1), z2) → c1(APP(z1, z2))
S tuples:

APP(add(z0, z1), z2) → c1(APP(z1, z2))
K tuples:none
Defined Rule Symbols:

reverse, app

Defined Pair Symbols:

APP, REVERSE, SHUFFLE

Compound Symbols:

c1, c3, c5

### (9) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

We considered the (Usable) Rules:

reverse(nil) → nil
app(nil, z0) → z0
And the Tuples:

APP(add(z0, z1), z2) → c1(APP(z1, z2))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(APP(x1, x2)) = 0
POL(REVERSE(x1)) = 0
POL(SHUFFLE(x1)) = x1
POL(add(x1, x2)) = [1] + x2
POL(app(x1, x2)) = x1 + x2
POL(c1(x1)) = x1
POL(c3(x1, x2)) = x1 + x2
POL(c5(x1, x2)) = x1 + x2
POL(nil) = 0
POL(reverse(x1)) = x1

### (10) Obligation:

Complexity Dependency Tuples Problem
Rules:

reverse(nil) → nil
app(nil, z0) → z0
Tuples:

APP(add(z0, z1), z2) → c1(APP(z1, z2))
S tuples:

APP(add(z0, z1), z2) → c1(APP(z1, z2))
K tuples:

Defined Rule Symbols:

reverse, app

Defined Pair Symbols:

APP, REVERSE, SHUFFLE

Compound Symbols:

c1, c3, c5

### (11) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

We considered the (Usable) Rules:

reverse(nil) → nil
app(nil, z0) → z0
And the Tuples:

APP(add(z0, z1), z2) → c1(APP(z1, z2))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(APP(x1, x2)) = 0
POL(REVERSE(x1)) = [1] + x1
POL(SHUFFLE(x1)) = [2]x12
POL(add(x1, x2)) = [1] + x1 + x2
POL(app(x1, x2)) = x1 + x2
POL(c1(x1)) = x1
POL(c3(x1, x2)) = x1 + x2
POL(c5(x1, x2)) = x1 + x2
POL(nil) = 0
POL(reverse(x1)) = x1

### (12) Obligation:

Complexity Dependency Tuples Problem
Rules:

reverse(nil) → nil
app(nil, z0) → z0
Tuples:

APP(add(z0, z1), z2) → c1(APP(z1, z2))
S tuples:

APP(add(z0, z1), z2) → c1(APP(z1, z2))
K tuples:

Defined Rule Symbols:

reverse, app

Defined Pair Symbols:

APP, REVERSE, SHUFFLE

Compound Symbols:

c1, c3, c5

### (13) CdtRuleRemovalProof (UPPER BOUND(ADD(n^3)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

APP(add(z0, z1), z2) → c1(APP(z1, z2))
We considered the (Usable) Rules:

reverse(nil) → nil
app(nil, z0) → z0
And the Tuples:

APP(add(z0, z1), z2) → c1(APP(z1, z2))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(APP(x1, x2)) = x1 + x1·x2
POL(REVERSE(x1)) = [1] + x12
POL(SHUFFLE(x1)) = x1 + x12 + x13
POL(add(x1, x2)) = [1] + x2
POL(app(x1, x2)) = x1 + x2
POL(c1(x1)) = x1
POL(c3(x1, x2)) = x1 + x2
POL(c5(x1, x2)) = x1 + x2
POL(nil) = 0
POL(reverse(x1)) = x1

### (14) Obligation:

Complexity Dependency Tuples Problem
Rules:

reverse(nil) → nil
app(nil, z0) → z0
Tuples:

APP(add(z0, z1), z2) → c1(APP(z1, z2))
S tuples:none
K tuples:

APP(add(z0, z1), z2) → c1(APP(z1, z2))
Defined Rule Symbols:

reverse, app

Defined Pair Symbols:

APP, REVERSE, SHUFFLE

Compound Symbols:

c1, c3, c5

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

The set S is empty