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

0 CpxTRS
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)
↳2 CdtProblem
↳3 CdtLeafRemovalProof (ComplexityIfPolyImplication, 0 ms)
↳4 CdtProblem
↳5 CdtUsableRulesProof (⇔, 0 ms)
↳6 CdtProblem
↳7 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 78 ms)
↳8 CdtProblem
↳9 SIsEmptyProof (BOTH BOUNDS(ID, ID), 0 ms)
↳10 BOUNDS(1, 1)

(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)

Removed 1 leading nodes:

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(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

(10) BOUNDS(1, 1)