### (0) Obligation:

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

duplicate(Cons(x, xs)) → Cons(x, Cons(x, duplicate(xs)))
duplicate(Nil) → Nil
goal(x) → duplicate(x)

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

### (2) Obligation:

Complexity Dependency Tuples Problem
Rules:

duplicate(Cons(z0, z1)) → Cons(z0, Cons(z0, duplicate(z1)))
duplicate(Nil) → Nil
goal(z0) → duplicate(z0)
Tuples:

DUPLICATE(Cons(z0, z1)) → c(DUPLICATE(z1))
DUPLICATE(Nil) → c1
GOAL(z0) → c2(DUPLICATE(z0))
S tuples:

DUPLICATE(Cons(z0, z1)) → c(DUPLICATE(z1))
DUPLICATE(Nil) → c1
GOAL(z0) → c2(DUPLICATE(z0))
K tuples:none
Defined Rule Symbols:

duplicate, goal

Defined Pair Symbols:

DUPLICATE, GOAL

Compound Symbols:

c, c1, c2

### (3) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

GOAL(z0) → c2(DUPLICATE(z0))
Removed 1 trailing nodes:

DUPLICATE(Nil) → c1

### (4) Obligation:

Complexity Dependency Tuples Problem
Rules:

duplicate(Cons(z0, z1)) → Cons(z0, Cons(z0, duplicate(z1)))
duplicate(Nil) → Nil
goal(z0) → duplicate(z0)
Tuples:

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

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

duplicate, goal

Defined Pair Symbols:

DUPLICATE

Compound Symbols:

c

### (5) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

duplicate(Cons(z0, z1)) → Cons(z0, Cons(z0, duplicate(z1)))
duplicate(Nil) → Nil
goal(z0) → duplicate(z0)

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

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

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

Defined Pair Symbols:

DUPLICATE

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.

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

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

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

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

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

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

Defined Pair Symbols:

DUPLICATE

Compound Symbols:

c

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

The set S is empty