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

(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

(18) BOUNDS(1, 1)