### (0) Obligation:

The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^2).

The TRS R consists of the following rules:

half(0) → 0
half(s(s(x))) → s(half(x))
log(s(0)) → 0
log(s(s(x))) → s(log(s(half(x))))

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^2).

The TRS R consists of the following rules:

half(0) → 0
half(s(s(x))) → s(half(x))
log(s(0)) → 0
log(s(s(x))) → s(log(s(half(x))))

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

### (4) Obligation:

Complexity Dependency Tuples Problem
Rules:

half(0) → 0
half(s(s(z0))) → s(half(z0))
log(s(0)) → 0
log(s(s(z0))) → s(log(s(half(z0))))
Tuples:

HALF(0) → c
HALF(s(s(z0))) → c1(HALF(z0))
LOG(s(0)) → c2
LOG(s(s(z0))) → c3(LOG(s(half(z0))), HALF(z0))
S tuples:

HALF(0) → c
HALF(s(s(z0))) → c1(HALF(z0))
LOG(s(0)) → c2
LOG(s(s(z0))) → c3(LOG(s(half(z0))), HALF(z0))
K tuples:none
Defined Rule Symbols:

half, log

Defined Pair Symbols:

HALF, LOG

Compound Symbols:

c, c1, c2, c3

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

Removed 2 trailing nodes:

LOG(s(0)) → c2
HALF(0) → c

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:

half(0) → 0
half(s(s(z0))) → s(half(z0))
log(s(0)) → 0
log(s(s(z0))) → s(log(s(half(z0))))
Tuples:

HALF(s(s(z0))) → c1(HALF(z0))
LOG(s(s(z0))) → c3(LOG(s(half(z0))), HALF(z0))
S tuples:

HALF(s(s(z0))) → c1(HALF(z0))
LOG(s(s(z0))) → c3(LOG(s(half(z0))), HALF(z0))
K tuples:none
Defined Rule Symbols:

half, log

Defined Pair Symbols:

HALF, LOG

Compound Symbols:

c1, c3

### (7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

log(s(0)) → 0
log(s(s(z0))) → s(log(s(half(z0))))

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:

half(0) → 0
half(s(s(z0))) → s(half(z0))
Tuples:

HALF(s(s(z0))) → c1(HALF(z0))
LOG(s(s(z0))) → c3(LOG(s(half(z0))), HALF(z0))
S tuples:

HALF(s(s(z0))) → c1(HALF(z0))
LOG(s(s(z0))) → c3(LOG(s(half(z0))), HALF(z0))
K tuples:none
Defined Rule Symbols:

half

Defined Pair Symbols:

HALF, LOG

Compound Symbols:

c1, c3

### (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.

LOG(s(s(z0))) → c3(LOG(s(half(z0))), HALF(z0))
We considered the (Usable) Rules:

half(s(s(z0))) → s(half(z0))
half(0) → 0
And the Tuples:

HALF(s(s(z0))) → c1(HALF(z0))
LOG(s(s(z0))) → c3(LOG(s(half(z0))), HALF(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(HALF(x1)) = 0
POL(LOG(x1)) = x1
POL(c1(x1)) = x1
POL(c3(x1, x2)) = x1 + x2
POL(half(x1)) = x1
POL(s(x1)) =  + x1

### (10) Obligation:

Complexity Dependency Tuples Problem
Rules:

half(0) → 0
half(s(s(z0))) → s(half(z0))
Tuples:

HALF(s(s(z0))) → c1(HALF(z0))
LOG(s(s(z0))) → c3(LOG(s(half(z0))), HALF(z0))
S tuples:

HALF(s(s(z0))) → c1(HALF(z0))
K tuples:

LOG(s(s(z0))) → c3(LOG(s(half(z0))), HALF(z0))
Defined Rule Symbols:

half

Defined Pair Symbols:

HALF, LOG

Compound Symbols:

c1, c3

### (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.

HALF(s(s(z0))) → c1(HALF(z0))
We considered the (Usable) Rules:

half(s(s(z0))) → s(half(z0))
half(0) → 0
And the Tuples:

HALF(s(s(z0))) → c1(HALF(z0))
LOG(s(s(z0))) → c3(LOG(s(half(z0))), HALF(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0
POL(HALF(x1)) = x1
POL(LOG(x1)) = x12
POL(c1(x1)) = x1
POL(c3(x1, x2)) = x1 + x2
POL(half(x1)) = x1
POL(s(x1)) =  + x1

### (12) Obligation:

Complexity Dependency Tuples Problem
Rules:

half(0) → 0
half(s(s(z0))) → s(half(z0))
Tuples:

HALF(s(s(z0))) → c1(HALF(z0))
LOG(s(s(z0))) → c3(LOG(s(half(z0))), HALF(z0))
S tuples:none
K tuples:

LOG(s(s(z0))) → c3(LOG(s(half(z0))), HALF(z0))
HALF(s(s(z0))) → c1(HALF(z0))
Defined Rule Symbols:

half

Defined Pair Symbols:

HALF, LOG

Compound Symbols:

c1, c3

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

The set S is empty