R
↳Dependency Pair Analysis
SUM(s(x)) -> SQR(s(x))
SUM(s(x)) -> SUM(x)
R
↳DPs
→DP Problem 1
↳Forward Instantiation Transformation
SUM(s(x)) -> SUM(x)
sum(0) -> 0
sum(s(x)) -> +(sqr(s(x)), sum(x))
sum(s(x)) -> +(*(s(x), s(x)), sum(x))
sqr(x) -> *(x, x)
one new Dependency Pair is created:
SUM(s(x)) -> SUM(x)
SUM(s(s(x''))) -> SUM(s(x''))
R
↳DPs
→DP Problem 1
↳FwdInst
→DP Problem 2
↳Forward Instantiation Transformation
SUM(s(s(x''))) -> SUM(s(x''))
sum(0) -> 0
sum(s(x)) -> +(sqr(s(x)), sum(x))
sum(s(x)) -> +(*(s(x), s(x)), sum(x))
sqr(x) -> *(x, x)
one new Dependency Pair is created:
SUM(s(s(x''))) -> SUM(s(x''))
SUM(s(s(s(x'''')))) -> SUM(s(s(x'''')))
R
↳DPs
→DP Problem 1
↳FwdInst
→DP Problem 2
↳FwdInst
...
→DP Problem 3
↳Argument Filtering and Ordering
SUM(s(s(s(x'''')))) -> SUM(s(s(x'''')))
sum(0) -> 0
sum(s(x)) -> +(sqr(s(x)), sum(x))
sum(s(x)) -> +(*(s(x), s(x)), sum(x))
sqr(x) -> *(x, x)
SUM(s(s(s(x'''')))) -> SUM(s(s(x'''')))
sum(0) -> 0
sum(s(x)) -> +(sqr(s(x)), sum(x))
sum(s(x)) -> +(*(s(x), s(x)), sum(x))
sqr(x) -> *(x, x)
POL(SUM(x1)) = x1 POL(0) = 0 POL(sqr) = 0 POL(*) = 0 POL(s(x1)) = 1 + x1 POL(sum(x1)) = x1 POL(+(x1, x2)) = x1 + x2
SUM(x1) -> SUM(x1)
s(x1) -> s(x1)
sum(x1) -> sum(x1)
+(x1, x2) -> +(x1, x2)
sqr(x1) -> sqr
*(x1, x2) -> *
R
↳DPs
→DP Problem 1
↳FwdInst
→DP Problem 2
↳FwdInst
...
→DP Problem 4
↳Dependency Graph
sum(0) -> 0
sum(s(x)) -> +(sqr(s(x)), sum(x))
sum(s(x)) -> +(*(s(x), s(x)), sum(x))
sqr(x) -> *(x, x)