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

0 CpxTRS
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)
↳2 CdtProblem
↳3 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
↳4 CdtProblem
↳5 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
↳6 CdtProblem
↳7 CdtUsableRulesProof (⇔, 0 ms)
↳8 CdtProblem
↳9 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))), 36 ms)
↳10 CdtProblem
↳11 SIsEmptyProof (BOTH BOUNDS(ID, ID), 0 ms)
↳12 BOUNDS(O(1), O(1))

(0) Obligation:

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

dbl(S(0), S(0)) → S(S(S(S(0))))
unsafe(S(x)) → dbl(unsafe(x), 0)
unsafe(0) → 0
dbl(0, y) → y

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

dbl(S(0), S(0)) → S(S(S(S(0))))
dbl(0, z0) → z0
unsafe(S(z0)) → dbl(unsafe(z0), 0)
unsafe(0) → 0
Tuples:

DBL(S(0), S(0)) → c
DBL(0, z0) → c1
UNSAFE(S(z0)) → c2(DBL(unsafe(z0), 0), UNSAFE(z0))
UNSAFE(0) → c3
S tuples:

DBL(S(0), S(0)) → c
DBL(0, z0) → c1
UNSAFE(S(z0)) → c2(DBL(unsafe(z0), 0), UNSAFE(z0))
UNSAFE(0) → c3
K tuples:none
Defined Rule Symbols:

dbl, unsafe

Defined Pair Symbols:

DBL, UNSAFE

Compound Symbols:

c, c1, c2, c3

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

Removed 3 trailing nodes:

UNSAFE(0) → c3
DBL(0, z0) → c1
DBL(S(0), S(0)) → c

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

dbl(S(0), S(0)) → S(S(S(S(0))))
dbl(0, z0) → z0
unsafe(S(z0)) → dbl(unsafe(z0), 0)
unsafe(0) → 0
Tuples:

UNSAFE(S(z0)) → c2(DBL(unsafe(z0), 0), UNSAFE(z0))
S tuples:

UNSAFE(S(z0)) → c2(DBL(unsafe(z0), 0), UNSAFE(z0))
K tuples:none
Defined Rule Symbols:

dbl, unsafe

Defined Pair Symbols:

UNSAFE

Compound Symbols:

c2

(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

dbl(S(0), S(0)) → S(S(S(S(0))))
dbl(0, z0) → z0
unsafe(S(z0)) → dbl(unsafe(z0), 0)
unsafe(0) → 0
Tuples:

UNSAFE(S(z0)) → c2(UNSAFE(z0))
S tuples:

UNSAFE(S(z0)) → c2(UNSAFE(z0))
K tuples:none
Defined Rule Symbols:

dbl, unsafe

Defined Pair Symbols:

UNSAFE

Compound Symbols:

c2

(7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

dbl(S(0), S(0)) → S(S(S(S(0))))
dbl(0, z0) → z0
unsafe(S(z0)) → dbl(unsafe(z0), 0)
unsafe(0) → 0

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

UNSAFE(S(z0)) → c2(UNSAFE(z0))
S tuples:

UNSAFE(S(z0)) → c2(UNSAFE(z0))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

UNSAFE

Compound Symbols:

c2

(9) 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.

UNSAFE(S(z0)) → c2(UNSAFE(z0))
We considered the (Usable) Rules:none
And the Tuples:

UNSAFE(S(z0)) → c2(UNSAFE(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(S(x1)) = [1] + x1   
POL(UNSAFE(x1)) = [5]x1   
POL(c2(x1)) = x1   

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

UNSAFE(S(z0)) → c2(UNSAFE(z0))
S tuples:none
K tuples:

UNSAFE(S(z0)) → c2(UNSAFE(z0))
Defined Rule Symbols:none

Defined Pair Symbols:

UNSAFE

Compound Symbols:

c2

(11) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

The set S is empty

(12) BOUNDS(O(1), O(1))