### (0) Obligation:

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

dbl(S(0), S(0)) → S(S(S(S(0))))
save(S(x)) → dbl(0, save(x))
save(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
save(S(z0)) → dbl(0, save(z0))
save(0) → 0
Tuples:

DBL(S(0), S(0)) → c
DBL(0, z0) → c1
SAVE(S(z0)) → c2(DBL(0, save(z0)), SAVE(z0))
SAVE(0) → c3
S tuples:

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

dbl, save

Defined Pair Symbols:

DBL, SAVE

Compound Symbols:

c, c1, c2, c3

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

Removed 3 trailing nodes:

SAVE(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
save(S(z0)) → dbl(0, save(z0))
save(0) → 0
Tuples:

SAVE(S(z0)) → c2(DBL(0, save(z0)), SAVE(z0))
S tuples:

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

dbl, save

Defined Pair Symbols:

SAVE

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

SAVE(S(z0)) → c2(SAVE(z0))
S tuples:

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

dbl, save

Defined Pair Symbols:

SAVE

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

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

SAVE(S(z0)) → c2(SAVE(z0))
S tuples:

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

Defined Pair Symbols:

SAVE

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.

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

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

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

### (10) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

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

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

Defined Pair Symbols:

SAVE

Compound Symbols:

c2

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

The set S is empty