R
↳Dependency Pair Analysis
2NDSPOS(s(N), cons(X, Z)) -> 2NDSPOS(s(N), cons2(X, activate(Z)))
2NDSPOS(s(N), cons(X, Z)) -> ACTIVATE(Z)
2NDSPOS(s(N), cons2(X, cons(Y, Z))) -> 2NDSNEG(N, activate(Z))
2NDSPOS(s(N), cons2(X, cons(Y, Z))) -> ACTIVATE(Z)
2NDSNEG(s(N), cons(X, Z)) -> 2NDSNEG(s(N), cons2(X, activate(Z)))
2NDSNEG(s(N), cons(X, Z)) -> ACTIVATE(Z)
2NDSNEG(s(N), cons2(X, cons(Y, Z))) -> 2NDSPOS(N, activate(Z))
2NDSNEG(s(N), cons2(X, cons(Y, Z))) -> ACTIVATE(Z)
PI(X) -> 2NDSPOS(X, from(0))
PI(X) -> FROM(0)
PLUS(s(X), Y) -> PLUS(X, Y)
TIMES(s(X), Y) -> PLUS(Y, times(X, Y))
TIMES(s(X), Y) -> TIMES(X, Y)
SQUARE(X) -> TIMES(X, X)
ACTIVATE(nfrom(X)) -> FROM(X)
R
↳DPs
→DP Problem 1
↳Narrowing Transformation
→DP Problem 2
↳FwdInst
→DP Problem 3
↳FwdInst
2NDSNEG(s(N), cons2(X, cons(Y, Z))) -> 2NDSPOS(N, activate(Z))
2NDSNEG(s(N), cons(X, Z)) -> 2NDSNEG(s(N), cons2(X, activate(Z)))
2NDSPOS(s(N), cons2(X, cons(Y, Z))) -> 2NDSNEG(N, activate(Z))
2NDSPOS(s(N), cons(X, Z)) -> 2NDSPOS(s(N), cons2(X, activate(Z)))
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
activate(nfrom(X)) -> from(X)
activate(X) -> X
innermost
two new Dependency Pairs are created:
2NDSPOS(s(N), cons(X, Z)) -> 2NDSPOS(s(N), cons2(X, activate(Z)))
2NDSPOS(s(N), cons(X, nfrom(X''))) -> 2NDSPOS(s(N), cons2(X, from(X'')))
2NDSPOS(s(N), cons(X, Z')) -> 2NDSPOS(s(N), cons2(X, Z'))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 4
↳Narrowing Transformation
→DP Problem 2
↳FwdInst
→DP Problem 3
↳FwdInst
2NDSPOS(s(N), cons(X, Z')) -> 2NDSPOS(s(N), cons2(X, Z'))
2NDSPOS(s(N), cons(X, nfrom(X''))) -> 2NDSPOS(s(N), cons2(X, from(X'')))
2NDSNEG(s(N), cons(X, Z)) -> 2NDSNEG(s(N), cons2(X, activate(Z)))
2NDSPOS(s(N), cons2(X, cons(Y, Z))) -> 2NDSNEG(N, activate(Z))
2NDSNEG(s(N), cons2(X, cons(Y, Z))) -> 2NDSPOS(N, activate(Z))
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
activate(nfrom(X)) -> from(X)
activate(X) -> X
innermost
two new Dependency Pairs are created:
2NDSPOS(s(N), cons2(X, cons(Y, Z))) -> 2NDSNEG(N, activate(Z))
2NDSPOS(s(N), cons2(X, cons(Y, nfrom(X'')))) -> 2NDSNEG(N, from(X''))
2NDSPOS(s(N), cons2(X, cons(Y, Z'))) -> 2NDSNEG(N, Z')
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 4
↳Nar
...
→DP Problem 5
↳Narrowing Transformation
→DP Problem 2
↳FwdInst
→DP Problem 3
↳FwdInst
2NDSPOS(s(N), cons2(X, cons(Y, Z'))) -> 2NDSNEG(N, Z')
2NDSPOS(s(N), cons(X, nfrom(X''))) -> 2NDSPOS(s(N), cons2(X, from(X'')))
2NDSNEG(s(N), cons2(X, cons(Y, Z))) -> 2NDSPOS(N, activate(Z))
2NDSNEG(s(N), cons(X, Z)) -> 2NDSNEG(s(N), cons2(X, activate(Z)))
2NDSPOS(s(N), cons2(X, cons(Y, nfrom(X'')))) -> 2NDSNEG(N, from(X''))
2NDSPOS(s(N), cons(X, Z')) -> 2NDSPOS(s(N), cons2(X, Z'))
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
activate(nfrom(X)) -> from(X)
activate(X) -> X
innermost
two new Dependency Pairs are created:
2NDSNEG(s(N), cons(X, Z)) -> 2NDSNEG(s(N), cons2(X, activate(Z)))
2NDSNEG(s(N), cons(X, nfrom(X''))) -> 2NDSNEG(s(N), cons2(X, from(X'')))
2NDSNEG(s(N), cons(X, Z')) -> 2NDSNEG(s(N), cons2(X, Z'))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 4
↳Nar
...
→DP Problem 6
↳Narrowing Transformation
→DP Problem 2
↳FwdInst
→DP Problem 3
↳FwdInst
2NDSPOS(s(N), cons(X, Z')) -> 2NDSPOS(s(N), cons2(X, Z'))
2NDSNEG(s(N), cons(X, Z')) -> 2NDSNEG(s(N), cons2(X, Z'))
2NDSNEG(s(N), cons(X, nfrom(X''))) -> 2NDSNEG(s(N), cons2(X, from(X'')))
2NDSPOS(s(N), cons2(X, cons(Y, nfrom(X'')))) -> 2NDSNEG(N, from(X''))
2NDSPOS(s(N), cons(X, nfrom(X''))) -> 2NDSPOS(s(N), cons2(X, from(X'')))
2NDSNEG(s(N), cons2(X, cons(Y, Z))) -> 2NDSPOS(N, activate(Z))
2NDSPOS(s(N), cons2(X, cons(Y, Z'))) -> 2NDSNEG(N, Z')
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
activate(nfrom(X)) -> from(X)
activate(X) -> X
innermost
two new Dependency Pairs are created:
2NDSNEG(s(N), cons2(X, cons(Y, Z))) -> 2NDSPOS(N, activate(Z))
2NDSNEG(s(N), cons2(X, cons(Y, nfrom(X'')))) -> 2NDSPOS(N, from(X''))
2NDSNEG(s(N), cons2(X, cons(Y, Z'))) -> 2NDSPOS(N, Z')
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 4
↳Nar
...
→DP Problem 7
↳Narrowing Transformation
→DP Problem 2
↳FwdInst
→DP Problem 3
↳FwdInst
2NDSNEG(s(N), cons2(X, cons(Y, Z'))) -> 2NDSPOS(N, Z')
2NDSNEG(s(N), cons(X, Z')) -> 2NDSNEG(s(N), cons2(X, Z'))
2NDSPOS(s(N), cons2(X, cons(Y, Z'))) -> 2NDSNEG(N, Z')
2NDSPOS(s(N), cons(X, nfrom(X''))) -> 2NDSPOS(s(N), cons2(X, from(X'')))
2NDSNEG(s(N), cons2(X, cons(Y, nfrom(X'')))) -> 2NDSPOS(N, from(X''))
2NDSNEG(s(N), cons(X, nfrom(X''))) -> 2NDSNEG(s(N), cons2(X, from(X'')))
2NDSPOS(s(N), cons2(X, cons(Y, nfrom(X'')))) -> 2NDSNEG(N, from(X''))
2NDSPOS(s(N), cons(X, Z')) -> 2NDSPOS(s(N), cons2(X, Z'))
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
activate(nfrom(X)) -> from(X)
activate(X) -> X
innermost
two new Dependency Pairs are created:
2NDSPOS(s(N), cons(X, nfrom(X''))) -> 2NDSPOS(s(N), cons2(X, from(X'')))
2NDSPOS(s(N), cons(X, nfrom(X'''))) -> 2NDSPOS(s(N), cons2(X, cons(X''', nfrom(s(X''')))))
2NDSPOS(s(N), cons(X, nfrom(X'''))) -> 2NDSPOS(s(N), cons2(X, nfrom(X''')))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 4
↳Nar
...
→DP Problem 8
↳Narrowing Transformation
→DP Problem 2
↳FwdInst
→DP Problem 3
↳FwdInst
2NDSPOS(s(N), cons(X, nfrom(X'''))) -> 2NDSPOS(s(N), cons2(X, cons(X''', nfrom(s(X''')))))
2NDSNEG(s(N), cons(X, Z')) -> 2NDSNEG(s(N), cons2(X, Z'))
2NDSPOS(s(N), cons2(X, cons(Y, Z'))) -> 2NDSNEG(N, Z')
2NDSNEG(s(N), cons2(X, cons(Y, nfrom(X'')))) -> 2NDSPOS(N, from(X''))
2NDSNEG(s(N), cons(X, nfrom(X''))) -> 2NDSNEG(s(N), cons2(X, from(X'')))
2NDSPOS(s(N), cons2(X, cons(Y, nfrom(X'')))) -> 2NDSNEG(N, from(X''))
2NDSPOS(s(N), cons(X, Z')) -> 2NDSPOS(s(N), cons2(X, Z'))
2NDSNEG(s(N), cons2(X, cons(Y, Z'))) -> 2NDSPOS(N, Z')
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
activate(nfrom(X)) -> from(X)
activate(X) -> X
innermost
two new Dependency Pairs are created:
2NDSPOS(s(N), cons2(X, cons(Y, nfrom(X'')))) -> 2NDSNEG(N, from(X''))
2NDSPOS(s(N), cons2(X, cons(Y, nfrom(X''')))) -> 2NDSNEG(N, cons(X''', nfrom(s(X'''))))
2NDSPOS(s(N), cons2(X, cons(Y, nfrom(X''')))) -> 2NDSNEG(N, nfrom(X'''))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 4
↳Nar
...
→DP Problem 9
↳Narrowing Transformation
→DP Problem 2
↳FwdInst
→DP Problem 3
↳FwdInst
2NDSNEG(s(N), cons2(X, cons(Y, Z'))) -> 2NDSPOS(N, Z')
2NDSNEG(s(N), cons(X, Z')) -> 2NDSNEG(s(N), cons2(X, Z'))
2NDSPOS(s(N), cons2(X, cons(Y, nfrom(X''')))) -> 2NDSNEG(N, cons(X''', nfrom(s(X'''))))
2NDSPOS(s(N), cons(X, Z')) -> 2NDSPOS(s(N), cons2(X, Z'))
2NDSNEG(s(N), cons2(X, cons(Y, nfrom(X'')))) -> 2NDSPOS(N, from(X''))
2NDSNEG(s(N), cons(X, nfrom(X''))) -> 2NDSNEG(s(N), cons2(X, from(X'')))
2NDSPOS(s(N), cons2(X, cons(Y, Z'))) -> 2NDSNEG(N, Z')
2NDSPOS(s(N), cons(X, nfrom(X'''))) -> 2NDSPOS(s(N), cons2(X, cons(X''', nfrom(s(X''')))))
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
activate(nfrom(X)) -> from(X)
activate(X) -> X
innermost
two new Dependency Pairs are created:
2NDSNEG(s(N), cons(X, nfrom(X''))) -> 2NDSNEG(s(N), cons2(X, from(X'')))
2NDSNEG(s(N), cons(X, nfrom(X'''))) -> 2NDSNEG(s(N), cons2(X, cons(X''', nfrom(s(X''')))))
2NDSNEG(s(N), cons(X, nfrom(X'''))) -> 2NDSNEG(s(N), cons2(X, nfrom(X''')))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 4
↳Nar
...
→DP Problem 10
↳Narrowing Transformation
→DP Problem 2
↳FwdInst
→DP Problem 3
↳FwdInst
2NDSNEG(s(N), cons(X, nfrom(X'''))) -> 2NDSNEG(s(N), cons2(X, cons(X''', nfrom(s(X''')))))
2NDSPOS(s(N), cons2(X, cons(Y, nfrom(X''')))) -> 2NDSNEG(N, cons(X''', nfrom(s(X'''))))
2NDSPOS(s(N), cons(X, nfrom(X'''))) -> 2NDSPOS(s(N), cons2(X, cons(X''', nfrom(s(X''')))))
2NDSNEG(s(N), cons2(X, cons(Y, nfrom(X'')))) -> 2NDSPOS(N, from(X''))
2NDSNEG(s(N), cons(X, Z')) -> 2NDSNEG(s(N), cons2(X, Z'))
2NDSPOS(s(N), cons2(X, cons(Y, Z'))) -> 2NDSNEG(N, Z')
2NDSPOS(s(N), cons(X, Z')) -> 2NDSPOS(s(N), cons2(X, Z'))
2NDSNEG(s(N), cons2(X, cons(Y, Z'))) -> 2NDSPOS(N, Z')
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
activate(nfrom(X)) -> from(X)
activate(X) -> X
innermost
two new Dependency Pairs are created:
2NDSNEG(s(N), cons2(X, cons(Y, nfrom(X'')))) -> 2NDSPOS(N, from(X''))
2NDSNEG(s(N), cons2(X, cons(Y, nfrom(X''')))) -> 2NDSPOS(N, cons(X''', nfrom(s(X'''))))
2NDSNEG(s(N), cons2(X, cons(Y, nfrom(X''')))) -> 2NDSPOS(N, nfrom(X'''))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 4
↳Nar
...
→DP Problem 11
↳Forward Instantiation Transformation
→DP Problem 2
↳FwdInst
→DP Problem 3
↳FwdInst
2NDSPOS(s(N), cons2(X, cons(Y, nfrom(X''')))) -> 2NDSNEG(N, cons(X''', nfrom(s(X'''))))
2NDSPOS(s(N), cons(X, nfrom(X'''))) -> 2NDSPOS(s(N), cons2(X, cons(X''', nfrom(s(X''')))))
2NDSNEG(s(N), cons2(X, cons(Y, nfrom(X''')))) -> 2NDSPOS(N, cons(X''', nfrom(s(X'''))))
2NDSNEG(s(N), cons(X, Z')) -> 2NDSNEG(s(N), cons2(X, Z'))
2NDSPOS(s(N), cons2(X, cons(Y, Z'))) -> 2NDSNEG(N, Z')
2NDSPOS(s(N), cons(X, Z')) -> 2NDSPOS(s(N), cons2(X, Z'))
2NDSNEG(s(N), cons2(X, cons(Y, Z'))) -> 2NDSPOS(N, Z')
2NDSNEG(s(N), cons(X, nfrom(X'''))) -> 2NDSNEG(s(N), cons2(X, cons(X''', nfrom(s(X''')))))
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
activate(nfrom(X)) -> from(X)
activate(X) -> X
innermost
two new Dependency Pairs are created:
2NDSPOS(s(N), cons(X, Z')) -> 2NDSPOS(s(N), cons2(X, Z'))
2NDSPOS(s(N''), cons(X'', cons(Y'', Z'''))) -> 2NDSPOS(s(N''), cons2(X'', cons(Y'', Z''')))
2NDSPOS(s(N''), cons(X'', cons(Y'', nfrom(X''''')))) -> 2NDSPOS(s(N''), cons2(X'', cons(Y'', nfrom(X'''''))))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 4
↳Nar
...
→DP Problem 12
↳Forward Instantiation Transformation
→DP Problem 2
↳FwdInst
→DP Problem 3
↳FwdInst
2NDSPOS(s(N''), cons(X'', cons(Y'', nfrom(X''''')))) -> 2NDSPOS(s(N''), cons2(X'', cons(Y'', nfrom(X'''''))))
2NDSPOS(s(N''), cons(X'', cons(Y'', Z'''))) -> 2NDSPOS(s(N''), cons2(X'', cons(Y'', Z''')))
2NDSPOS(s(N), cons(X, nfrom(X'''))) -> 2NDSPOS(s(N), cons2(X, cons(X''', nfrom(s(X''')))))
2NDSNEG(s(N), cons2(X, cons(Y, nfrom(X''')))) -> 2NDSPOS(N, cons(X''', nfrom(s(X'''))))
2NDSNEG(s(N), cons(X, nfrom(X'''))) -> 2NDSNEG(s(N), cons2(X, cons(X''', nfrom(s(X''')))))
2NDSPOS(s(N), cons2(X, cons(Y, Z'))) -> 2NDSNEG(N, Z')
2NDSNEG(s(N), cons2(X, cons(Y, Z'))) -> 2NDSPOS(N, Z')
2NDSNEG(s(N), cons(X, Z')) -> 2NDSNEG(s(N), cons2(X, Z'))
2NDSPOS(s(N), cons2(X, cons(Y, nfrom(X''')))) -> 2NDSNEG(N, cons(X''', nfrom(s(X'''))))
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
activate(nfrom(X)) -> from(X)
activate(X) -> X
innermost
four new Dependency Pairs are created:
2NDSPOS(s(N), cons2(X, cons(Y, Z'))) -> 2NDSNEG(N, Z')
2NDSPOS(s(s(N'')), cons2(X, cons(Y, cons(X'', Z''')))) -> 2NDSNEG(s(N''), cons(X'', Z'''))
2NDSPOS(s(s(N'')), cons2(X, cons(Y, cons2(X'', cons(Y'', Z'''))))) -> 2NDSNEG(s(N''), cons2(X'', cons(Y'', Z''')))
2NDSPOS(s(s(N'')), cons2(X, cons(Y, cons(X'', nfrom(X'''''))))) -> 2NDSNEG(s(N''), cons(X'', nfrom(X''''')))
2NDSPOS(s(s(N'')), cons2(X, cons(Y, cons2(X'', cons(Y'', nfrom(X''''')))))) -> 2NDSNEG(s(N''), cons2(X'', cons(Y'', nfrom(X'''''))))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 4
↳Nar
...
→DP Problem 13
↳Forward Instantiation Transformation
→DP Problem 2
↳FwdInst
→DP Problem 3
↳FwdInst
2NDSPOS(s(s(N'')), cons2(X, cons(Y, cons2(X'', cons(Y'', nfrom(X''''')))))) -> 2NDSNEG(s(N''), cons2(X'', cons(Y'', nfrom(X'''''))))
2NDSPOS(s(s(N'')), cons2(X, cons(Y, cons(X'', nfrom(X'''''))))) -> 2NDSNEG(s(N''), cons(X'', nfrom(X''''')))
2NDSPOS(s(s(N'')), cons2(X, cons(Y, cons2(X'', cons(Y'', Z'''))))) -> 2NDSNEG(s(N''), cons2(X'', cons(Y'', Z''')))
2NDSNEG(s(N), cons2(X, cons(Y, nfrom(X''')))) -> 2NDSPOS(N, cons(X''', nfrom(s(X'''))))
2NDSNEG(s(N), cons(X, nfrom(X'''))) -> 2NDSNEG(s(N), cons2(X, cons(X''', nfrom(s(X''')))))
2NDSPOS(s(s(N'')), cons2(X, cons(Y, cons(X'', Z''')))) -> 2NDSNEG(s(N''), cons(X'', Z'''))
2NDSPOS(s(N''), cons(X'', cons(Y'', Z'''))) -> 2NDSPOS(s(N''), cons2(X'', cons(Y'', Z''')))
2NDSPOS(s(N), cons(X, nfrom(X'''))) -> 2NDSPOS(s(N), cons2(X, cons(X''', nfrom(s(X''')))))
2NDSNEG(s(N), cons2(X, cons(Y, Z'))) -> 2NDSPOS(N, Z')
2NDSNEG(s(N), cons(X, Z')) -> 2NDSNEG(s(N), cons2(X, Z'))
2NDSPOS(s(N), cons2(X, cons(Y, nfrom(X''')))) -> 2NDSNEG(N, cons(X''', nfrom(s(X'''))))
2NDSPOS(s(N''), cons(X'', cons(Y'', nfrom(X''''')))) -> 2NDSPOS(s(N''), cons2(X'', cons(Y'', nfrom(X'''''))))
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
activate(nfrom(X)) -> from(X)
activate(X) -> X
innermost
two new Dependency Pairs are created:
2NDSNEG(s(N), cons(X, Z')) -> 2NDSNEG(s(N), cons2(X, Z'))
2NDSNEG(s(N''), cons(X'', cons(Y'', Z'''))) -> 2NDSNEG(s(N''), cons2(X'', cons(Y'', Z''')))
2NDSNEG(s(N''), cons(X'', cons(Y'', nfrom(X''''')))) -> 2NDSNEG(s(N''), cons2(X'', cons(Y'', nfrom(X'''''))))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 4
↳Nar
...
→DP Problem 14
↳Forward Instantiation Transformation
→DP Problem 2
↳FwdInst
→DP Problem 3
↳FwdInst
2NDSPOS(s(N''), cons(X'', cons(Y'', nfrom(X''''')))) -> 2NDSPOS(s(N''), cons2(X'', cons(Y'', nfrom(X'''''))))
2NDSPOS(s(s(N'')), cons2(X, cons(Y, cons(X'', nfrom(X'''''))))) -> 2NDSNEG(s(N''), cons(X'', nfrom(X''''')))
2NDSPOS(s(s(N'')), cons2(X, cons(Y, cons2(X'', cons(Y'', Z'''))))) -> 2NDSNEG(s(N''), cons2(X'', cons(Y'', Z''')))
2NDSNEG(s(N''), cons(X'', cons(Y'', nfrom(X''''')))) -> 2NDSNEG(s(N''), cons2(X'', cons(Y'', nfrom(X'''''))))
2NDSNEG(s(N''), cons(X'', cons(Y'', Z'''))) -> 2NDSNEG(s(N''), cons2(X'', cons(Y'', Z''')))
2NDSPOS(s(s(N'')), cons2(X, cons(Y, cons(X'', Z''')))) -> 2NDSNEG(s(N''), cons(X'', Z'''))
2NDSPOS(s(N''), cons(X'', cons(Y'', Z'''))) -> 2NDSPOS(s(N''), cons2(X'', cons(Y'', Z''')))
2NDSNEG(s(N), cons2(X, cons(Y, nfrom(X''')))) -> 2NDSPOS(N, cons(X''', nfrom(s(X'''))))
2NDSNEG(s(N), cons(X, nfrom(X'''))) -> 2NDSNEG(s(N), cons2(X, cons(X''', nfrom(s(X''')))))
2NDSPOS(s(N), cons2(X, cons(Y, nfrom(X''')))) -> 2NDSNEG(N, cons(X''', nfrom(s(X'''))))
2NDSPOS(s(N), cons(X, nfrom(X'''))) -> 2NDSPOS(s(N), cons2(X, cons(X''', nfrom(s(X''')))))
2NDSNEG(s(N), cons2(X, cons(Y, Z'))) -> 2NDSPOS(N, Z')
2NDSPOS(s(s(N'')), cons2(X, cons(Y, cons2(X'', cons(Y'', nfrom(X''''')))))) -> 2NDSNEG(s(N''), cons2(X'', cons(Y'', nfrom(X'''''))))
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
activate(nfrom(X)) -> from(X)
activate(X) -> X
innermost
eight new Dependency Pairs are created:
2NDSNEG(s(N), cons2(X, cons(Y, Z'))) -> 2NDSPOS(N, Z')
2NDSNEG(s(s(N'')), cons2(X, cons(Y, cons(X'', nfrom(X'''''))))) -> 2NDSPOS(s(N''), cons(X'', nfrom(X''''')))
2NDSNEG(s(s(N'')), cons2(X, cons(Y, cons2(X'', cons(Y'', nfrom(X''''')))))) -> 2NDSPOS(s(N''), cons2(X'', cons(Y'', nfrom(X'''''))))
2NDSNEG(s(s(N'''')), cons2(X, cons(Y, cons(X'''', cons(Y'''', Z'''''))))) -> 2NDSPOS(s(N''''), cons(X'''', cons(Y'''', Z''''')))
2NDSNEG(s(s(N'''')), cons2(X, cons(Y, cons(X'''', cons(Y'''', nfrom(X''''''')))))) -> 2NDSPOS(s(N''''), cons(X'''', cons(Y'''', nfrom(X'''''''))))
2NDSNEG(s(s(s(N''''))), cons2(X, cons(Y, cons2(X'', cons(Y'', cons(X'''', Z''''')))))) -> 2NDSPOS(s(s(N'''')), cons2(X'', cons(Y'', cons(X'''', Z'''''))))
2NDSNEG(s(s(s(N''''))), cons2(X, cons(Y, cons2(X'', cons(Y'', cons2(X'''', cons(Y'''', Z'''''))))))) -> 2NDSPOS(s(s(N'''')), cons2(X'', cons(Y'', cons2(X'''', cons(Y'''', Z''''')))))
2NDSNEG(s(s(s(N''''))), cons2(X, cons(Y, cons2(X'', cons(Y'', cons(X'''', nfrom(X'''''''))))))) -> 2NDSPOS(s(s(N'''')), cons2(X'', cons(Y'', cons(X'''', nfrom(X''''''')))))
2NDSNEG(s(s(s(N''''))), cons2(X, cons(Y, cons2(X'', cons(Y'', cons2(X'''', cons(Y'''', nfrom(X''''''')))))))) -> 2NDSPOS(s(s(N'''')), cons2(X'', cons(Y'', cons2(X'''', cons(Y'''', nfrom(X'''''''))))))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 4
↳Nar
...
→DP Problem 15
↳Polynomial Ordering
→DP Problem 2
↳FwdInst
→DP Problem 3
↳FwdInst
2NDSPOS(s(N), cons(X, nfrom(X'''))) -> 2NDSPOS(s(N), cons2(X, cons(X''', nfrom(s(X''')))))
2NDSNEG(s(N), cons2(X, cons(Y, nfrom(X''')))) -> 2NDSPOS(N, cons(X''', nfrom(s(X'''))))
2NDSNEG(s(N), cons(X, nfrom(X'''))) -> 2NDSNEG(s(N), cons2(X, cons(X''', nfrom(s(X''')))))
2NDSPOS(s(N), cons2(X, cons(Y, nfrom(X''')))) -> 2NDSNEG(N, cons(X''', nfrom(s(X'''))))
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
activate(nfrom(X)) -> from(X)
activate(X) -> X
innermost
2NDSNEG(s(N), cons2(X, cons(Y, nfrom(X''')))) -> 2NDSPOS(N, cons(X''', nfrom(s(X'''))))
2NDSPOS(s(N), cons2(X, cons(Y, nfrom(X''')))) -> 2NDSNEG(N, cons(X''', nfrom(s(X'''))))
POL(n__from(x1)) = 0 POL(cons(x1, x2)) = 0 POL(2NDSNEG(x1, x2)) = x1 POL(s(x1)) = 1 + x1 POL(cons2(x1, x2)) = 0 POL(2NDSPOS(x1, x2)) = x1
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 4
↳Nar
...
→DP Problem 17
↳Dependency Graph
→DP Problem 2
↳FwdInst
→DP Problem 3
↳FwdInst
2NDSPOS(s(N), cons(X, nfrom(X'''))) -> 2NDSPOS(s(N), cons2(X, cons(X''', nfrom(s(X''')))))
2NDSNEG(s(N), cons(X, nfrom(X'''))) -> 2NDSNEG(s(N), cons2(X, cons(X''', nfrom(s(X''')))))
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
activate(nfrom(X)) -> from(X)
activate(X) -> X
innermost
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 4
↳Nar
...
→DP Problem 16
↳Polynomial Ordering
→DP Problem 2
↳FwdInst
→DP Problem 3
↳FwdInst
2NDSNEG(s(s(s(N''''))), cons2(X, cons(Y, cons2(X'', cons(Y'', cons2(X'''', cons(Y'''', nfrom(X''''''')))))))) -> 2NDSPOS(s(s(N'''')), cons2(X'', cons(Y'', cons2(X'''', cons(Y'''', nfrom(X'''''''))))))
2NDSNEG(s(s(s(N''''))), cons2(X, cons(Y, cons2(X'', cons(Y'', cons(X'''', nfrom(X'''''''))))))) -> 2NDSPOS(s(s(N'''')), cons2(X'', cons(Y'', cons(X'''', nfrom(X''''''')))))
2NDSNEG(s(s(s(N''''))), cons2(X, cons(Y, cons2(X'', cons(Y'', cons2(X'''', cons(Y'''', Z'''''))))))) -> 2NDSPOS(s(s(N'''')), cons2(X'', cons(Y'', cons2(X'''', cons(Y'''', Z''''')))))
2NDSNEG(s(s(s(N''''))), cons2(X, cons(Y, cons2(X'', cons(Y'', cons(X'''', Z''''')))))) -> 2NDSPOS(s(s(N'''')), cons2(X'', cons(Y'', cons(X'''', Z'''''))))
2NDSNEG(s(s(N'''')), cons2(X, cons(Y, cons(X'''', cons(Y'''', nfrom(X''''''')))))) -> 2NDSPOS(s(N''''), cons(X'''', cons(Y'''', nfrom(X'''''''))))
2NDSNEG(s(N''), cons(X'', cons(Y'', Z'''))) -> 2NDSNEG(s(N''), cons2(X'', cons(Y'', Z''')))
2NDSPOS(s(s(N'')), cons2(X, cons(Y, cons(X'', Z''')))) -> 2NDSNEG(s(N''), cons(X'', Z'''))
2NDSPOS(s(N''), cons(X'', cons(Y'', Z'''))) -> 2NDSPOS(s(N''), cons2(X'', cons(Y'', Z''')))
2NDSNEG(s(s(N'''')), cons2(X, cons(Y, cons(X'''', cons(Y'''', Z'''''))))) -> 2NDSPOS(s(N''''), cons(X'''', cons(Y'''', Z''''')))
2NDSPOS(s(s(N'')), cons2(X, cons(Y, cons2(X'', cons(Y'', Z'''))))) -> 2NDSNEG(s(N''), cons2(X'', cons(Y'', Z''')))
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
activate(nfrom(X)) -> from(X)
activate(X) -> X
innermost
2NDSNEG(s(s(s(N''''))), cons2(X, cons(Y, cons2(X'', cons(Y'', cons2(X'''', cons(Y'''', nfrom(X''''''')))))))) -> 2NDSPOS(s(s(N'''')), cons2(X'', cons(Y'', cons2(X'''', cons(Y'''', nfrom(X'''''''))))))
2NDSNEG(s(s(s(N''''))), cons2(X, cons(Y, cons2(X'', cons(Y'', cons(X'''', nfrom(X'''''''))))))) -> 2NDSPOS(s(s(N'''')), cons2(X'', cons(Y'', cons(X'''', nfrom(X''''''')))))
2NDSNEG(s(s(s(N''''))), cons2(X, cons(Y, cons2(X'', cons(Y'', cons2(X'''', cons(Y'''', Z'''''))))))) -> 2NDSPOS(s(s(N'''')), cons2(X'', cons(Y'', cons2(X'''', cons(Y'''', Z''''')))))
2NDSNEG(s(s(s(N''''))), cons2(X, cons(Y, cons2(X'', cons(Y'', cons(X'''', Z''''')))))) -> 2NDSPOS(s(s(N'''')), cons2(X'', cons(Y'', cons(X'''', Z'''''))))
2NDSNEG(s(s(N'''')), cons2(X, cons(Y, cons(X'''', cons(Y'''', nfrom(X''''''')))))) -> 2NDSPOS(s(N''''), cons(X'''', cons(Y'''', nfrom(X'''''''))))
2NDSPOS(s(s(N'')), cons2(X, cons(Y, cons(X'', Z''')))) -> 2NDSNEG(s(N''), cons(X'', Z'''))
2NDSNEG(s(s(N'''')), cons2(X, cons(Y, cons(X'''', cons(Y'''', Z'''''))))) -> 2NDSPOS(s(N''''), cons(X'''', cons(Y'''', Z''''')))
2NDSPOS(s(s(N'')), cons2(X, cons(Y, cons2(X'', cons(Y'', Z'''))))) -> 2NDSNEG(s(N''), cons2(X'', cons(Y'', Z''')))
POL(n__from(x1)) = 0 POL(cons(x1, x2)) = 0 POL(2NDSNEG(x1, x2)) = x1 POL(s(x1)) = 1 + x1 POL(cons2(x1, x2)) = 0 POL(2NDSPOS(x1, x2)) = x1
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 4
↳Nar
...
→DP Problem 18
↳Dependency Graph
→DP Problem 2
↳FwdInst
→DP Problem 3
↳FwdInst
2NDSNEG(s(N''), cons(X'', cons(Y'', Z'''))) -> 2NDSNEG(s(N''), cons2(X'', cons(Y'', Z''')))
2NDSPOS(s(N''), cons(X'', cons(Y'', Z'''))) -> 2NDSPOS(s(N''), cons2(X'', cons(Y'', Z''')))
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
activate(nfrom(X)) -> from(X)
activate(X) -> X
innermost
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Forward Instantiation Transformation
→DP Problem 3
↳FwdInst
PLUS(s(X), Y) -> PLUS(X, Y)
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
activate(nfrom(X)) -> from(X)
activate(X) -> X
innermost
one new Dependency Pair is created:
PLUS(s(X), Y) -> PLUS(X, Y)
PLUS(s(s(X'')), Y'') -> PLUS(s(X''), Y'')
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳FwdInst
→DP Problem 19
↳Forward Instantiation Transformation
→DP Problem 3
↳FwdInst
PLUS(s(s(X'')), Y'') -> PLUS(s(X''), Y'')
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
activate(nfrom(X)) -> from(X)
activate(X) -> X
innermost
one new Dependency Pair is created:
PLUS(s(s(X'')), Y'') -> PLUS(s(X''), Y'')
PLUS(s(s(s(X''''))), Y'''') -> PLUS(s(s(X'''')), Y'''')
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳FwdInst
→DP Problem 19
↳FwdInst
...
→DP Problem 20
↳Polynomial Ordering
→DP Problem 3
↳FwdInst
PLUS(s(s(s(X''''))), Y'''') -> PLUS(s(s(X'''')), Y'''')
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
activate(nfrom(X)) -> from(X)
activate(X) -> X
innermost
PLUS(s(s(s(X''''))), Y'''') -> PLUS(s(s(X'''')), Y'''')
POL(PLUS(x1, x2)) = 1 + x1 POL(s(x1)) = 1 + x1
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳FwdInst
→DP Problem 19
↳FwdInst
...
→DP Problem 21
↳Dependency Graph
→DP Problem 3
↳FwdInst
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
activate(nfrom(X)) -> from(X)
activate(X) -> X
innermost
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳FwdInst
→DP Problem 3
↳Forward Instantiation Transformation
TIMES(s(X), Y) -> TIMES(X, Y)
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
activate(nfrom(X)) -> from(X)
activate(X) -> X
innermost
one new Dependency Pair is created:
TIMES(s(X), Y) -> TIMES(X, Y)
TIMES(s(s(X'')), Y'') -> TIMES(s(X''), Y'')
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳FwdInst
→DP Problem 3
↳FwdInst
→DP Problem 22
↳Forward Instantiation Transformation
TIMES(s(s(X'')), Y'') -> TIMES(s(X''), Y'')
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
activate(nfrom(X)) -> from(X)
activate(X) -> X
innermost
one new Dependency Pair is created:
TIMES(s(s(X'')), Y'') -> TIMES(s(X''), Y'')
TIMES(s(s(s(X''''))), Y'''') -> TIMES(s(s(X'''')), Y'''')
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳FwdInst
→DP Problem 3
↳FwdInst
→DP Problem 22
↳FwdInst
...
→DP Problem 23
↳Polynomial Ordering
TIMES(s(s(s(X''''))), Y'''') -> TIMES(s(s(X'''')), Y'''')
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
activate(nfrom(X)) -> from(X)
activate(X) -> X
innermost
TIMES(s(s(s(X''''))), Y'''') -> TIMES(s(s(X'''')), Y'''')
POL(TIMES(x1, x2)) = 1 + x1 POL(s(x1)) = 1 + x1
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳FwdInst
→DP Problem 3
↳FwdInst
→DP Problem 22
↳FwdInst
...
→DP Problem 24
↳Dependency Graph
from(X) -> cons(X, nfrom(s(X)))
from(X) -> nfrom(X)
2ndspos(0, Z) -> rnil
2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) -> rnil
2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) -> 2ndspos(X, from(0))
plus(0, Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
times(0, Y) -> 0
times(s(X), Y) -> plus(Y, times(X, Y))
square(X) -> times(X, X)
activate(nfrom(X)) -> from(X)
activate(X) -> X
innermost