### (0) Obligation:

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

The TRS R consists of the following rules:

p(s(x)) → x
fac(0) → s(0)
fac(s(x)) → times(s(x), fac(p(s(x))))

Rewrite Strategy: FULL

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

Converted rc-obligation to irc-obligation.

The duplicating contexts are:
fac(s([]))

The defined contexts are:
fac([])
p(s([]))

[] just represents basic- or constructor-terms in the following defined contexts:
fac([])

As the TRS is an overlay system and the defined contexts and the duplicating contexts don't overlap, we have rc = irc.

### (2) Obligation:

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

The TRS R consists of the following rules:

p(s(x)) → x
fac(0) → s(0)
fac(s(x)) → times(s(x), fac(p(s(x))))

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

### (4) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(s(z0)) → z0
fac(0) → s(0)
fac(s(z0)) → times(s(z0), fac(p(s(z0))))
Tuples:

P(s(z0)) → c
FAC(0) → c1
FAC(s(z0)) → c2(FAC(p(s(z0))), P(s(z0)))
S tuples:

P(s(z0)) → c
FAC(0) → c1
FAC(s(z0)) → c2(FAC(p(s(z0))), P(s(z0)))
K tuples:none
Defined Rule Symbols:

p, fac

Defined Pair Symbols:

P, FAC

Compound Symbols:

c, c1, c2

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

Removed 2 trailing nodes:

FAC(0) → c1
P(s(z0)) → c

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(s(z0)) → z0
fac(0) → s(0)
fac(s(z0)) → times(s(z0), fac(p(s(z0))))
Tuples:

FAC(s(z0)) → c2(FAC(p(s(z0))), P(s(z0)))
S tuples:

FAC(s(z0)) → c2(FAC(p(s(z0))), P(s(z0)))
K tuples:none
Defined Rule Symbols:

p, fac

Defined Pair Symbols:

FAC

Compound Symbols:

c2

### (7) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing tuple parts

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(s(z0)) → z0
fac(0) → s(0)
fac(s(z0)) → times(s(z0), fac(p(s(z0))))
Tuples:

FAC(s(z0)) → c2(FAC(p(s(z0))))
S tuples:

FAC(s(z0)) → c2(FAC(p(s(z0))))
K tuples:none
Defined Rule Symbols:

p, fac

Defined Pair Symbols:

FAC

Compound Symbols:

c2

### (9) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

fac(0) → s(0)
fac(s(z0)) → times(s(z0), fac(p(s(z0))))

### (10) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(s(z0)) → z0
Tuples:

FAC(s(z0)) → c2(FAC(p(s(z0))))
S tuples:

FAC(s(z0)) → c2(FAC(p(s(z0))))
K tuples:none
Defined Rule Symbols:

p

Defined Pair Symbols:

FAC

Compound Symbols:

c2

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

Use narrowing to replace FAC(s(z0)) → c2(FAC(p(s(z0)))) by

FAC(s(z0)) → c2(FAC(z0))

### (12) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(s(z0)) → z0
Tuples:

FAC(s(z0)) → c2(FAC(z0))
S tuples:

FAC(s(z0)) → c2(FAC(z0))
K tuples:none
Defined Rule Symbols:

p

Defined Pair Symbols:

FAC

Compound Symbols:

c2

### (13) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

p(s(z0)) → z0

### (14) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

FAC(s(z0)) → c2(FAC(z0))
S tuples:

FAC(s(z0)) → c2(FAC(z0))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

FAC

Compound Symbols:

c2

### (15) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

FAC(s(z0)) → c2(FAC(z0))
We considered the (Usable) Rules:none
And the Tuples:

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

POL(FAC(x1)) = x1
POL(c2(x1)) = x1
POL(s(x1)) = [1] + x1

### (16) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

FAC(s(z0)) → c2(FAC(z0))
S tuples:none
K tuples:

FAC(s(z0)) → c2(FAC(z0))
Defined Rule Symbols:none

Defined Pair Symbols:

FAC

Compound Symbols:

c2

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

The set S is empty