(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
sum(0) → 0
sum(s(x)) → +(sqr(s(x)), sum(x))
sqr(x) → *(x, x)
sum(s(x)) → +(*(s(x), s(x)), sum(x))
Q is empty.
(1) Overlay + Local Confluence (EQUIVALENT transformation)
The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
sum(0) → 0
sum(s(x)) → +(sqr(s(x)), sum(x))
sqr(x) → *(x, x)
sum(s(x)) → +(*(s(x), s(x)), sum(x))
The set Q consists of the following terms:
sum(0)
sum(s(x0))
sqr(x0)
(3) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
SUM(s(x)) → SQR(s(x))
SUM(s(x)) → SUM(x)
The TRS R consists of the following rules:
sum(0) → 0
sum(s(x)) → +(sqr(s(x)), sum(x))
sqr(x) → *(x, x)
sum(s(x)) → +(*(s(x), s(x)), sum(x))
The set Q consists of the following terms:
sum(0)
sum(s(x0))
sqr(x0)
We have to consider all minimal (P,Q,R)-chains.
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
SUM(s(x)) → SUM(x)
The TRS R consists of the following rules:
sum(0) → 0
sum(s(x)) → +(sqr(s(x)), sum(x))
sqr(x) → *(x, x)
sum(s(x)) → +(*(s(x), s(x)), sum(x))
The set Q consists of the following terms:
sum(0)
sum(s(x0))
sqr(x0)
We have to consider all minimal (P,Q,R)-chains.
(7) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
SUM(s(x)) → SUM(x)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
SUM(
x1) =
x1
s(
x1) =
s(
x1)
sum(
x1) =
sum
0 =
0
+(
x1,
x2) =
x1
sqr(
x1) =
sqr
*(
x1,
x2) =
*
Lexicographic path order with status [LPO].
Quasi-Precedence:
[sum, sqr] > 0 > s1
[sum, sqr] > * > s1
Status:
*: []
sqr: []
s1: [1]
sum: []
0: []
The following usable rules [FROCOS05] were oriented:
sum(0) → 0
sum(s(x)) → +(sqr(s(x)), sum(x))
sqr(x) → *(x, x)
sum(s(x)) → +(*(s(x), s(x)), sum(x))
(8) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
sum(0) → 0
sum(s(x)) → +(sqr(s(x)), sum(x))
sqr(x) → *(x, x)
sum(s(x)) → +(*(s(x), s(x)), sum(x))
The set Q consists of the following terms:
sum(0)
sum(s(x0))
sqr(x0)
We have to consider all minimal (P,Q,R)-chains.
(9) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(10) TRUE