R
↳Dependency Pair Analysis
ACKIN(s(X), 0) -> U11(ackin(X, s(0)))
ACKIN(s(X), 0) -> ACKIN(X, s(0))
ACKIN(s(X), s(Y)) -> U21(ackin(s(X), Y), X)
ACKIN(s(X), s(Y)) -> ACKIN(s(X), Y)
U21(ackout(X), Y) -> U22(ackin(Y, X))
U21(ackout(X), Y) -> ACKIN(Y, X)
R
↳DPs
→DP Problem 1
↳Narrowing Transformation
ACKIN(s(X), s(Y)) -> ACKIN(s(X), Y)
U21(ackout(X), Y) -> ACKIN(Y, X)
ACKIN(s(X), s(Y)) -> U21(ackin(s(X), Y), X)
ACKIN(s(X), 0) -> ACKIN(X, s(0))
ackin(0, X) -> ackout(s(X))
ackin(s(X), 0) -> u11(ackin(X, s(0)))
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u11(ackout(X)) -> ackout(X)
u21(ackout(X), Y) -> u22(ackin(Y, X))
u22(ackout(X)) -> ackout(X)
innermost
two new Dependency Pairs are created:
ACKIN(s(X), s(Y)) -> U21(ackin(s(X), Y), X)
ACKIN(s(X''), s(0)) -> U21(u11(ackin(X'', s(0))), X'')
ACKIN(s(X''), s(s(Y''))) -> U21(u21(ackin(s(X''), Y''), X''), X'')
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Narrowing Transformation
ACKIN(s(X''), s(s(Y''))) -> U21(u21(ackin(s(X''), Y''), X''), X'')
U21(ackout(X), Y) -> ACKIN(Y, X)
ACKIN(s(X''), s(0)) -> U21(u11(ackin(X'', s(0))), X'')
ACKIN(s(X), 0) -> ACKIN(X, s(0))
ACKIN(s(X), s(Y)) -> ACKIN(s(X), Y)
ackin(0, X) -> ackout(s(X))
ackin(s(X), 0) -> u11(ackin(X, s(0)))
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u11(ackout(X)) -> ackout(X)
u21(ackout(X), Y) -> u22(ackin(Y, X))
u22(ackout(X)) -> ackout(X)
innermost
two new Dependency Pairs are created:
ACKIN(s(X''), s(0)) -> U21(u11(ackin(X'', s(0))), X'')
ACKIN(s(0), s(0)) -> U21(u11(ackout(s(s(0)))), 0)
ACKIN(s(s(X')), s(0)) -> U21(u11(u21(ackin(s(X'), 0), X')), s(X'))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 3
↳Rewriting Transformation
ACKIN(s(s(X')), s(0)) -> U21(u11(u21(ackin(s(X'), 0), X')), s(X'))
ACKIN(s(0), s(0)) -> U21(u11(ackout(s(s(0)))), 0)
ACKIN(s(X), s(Y)) -> ACKIN(s(X), Y)
ACKIN(s(X), 0) -> ACKIN(X, s(0))
U21(ackout(X), Y) -> ACKIN(Y, X)
ACKIN(s(X''), s(s(Y''))) -> U21(u21(ackin(s(X''), Y''), X''), X'')
ackin(0, X) -> ackout(s(X))
ackin(s(X), 0) -> u11(ackin(X, s(0)))
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u11(ackout(X)) -> ackout(X)
u21(ackout(X), Y) -> u22(ackin(Y, X))
u22(ackout(X)) -> ackout(X)
innermost
one new Dependency Pair is created:
ACKIN(s(0), s(0)) -> U21(u11(ackout(s(s(0)))), 0)
ACKIN(s(0), s(0)) -> U21(ackout(s(s(0))), 0)
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 4
↳Rewriting Transformation
ACKIN(s(0), s(0)) -> U21(ackout(s(s(0))), 0)
ACKIN(s(X''), s(s(Y''))) -> U21(u21(ackin(s(X''), Y''), X''), X'')
ACKIN(s(X), s(Y)) -> ACKIN(s(X), Y)
ACKIN(s(X), 0) -> ACKIN(X, s(0))
U21(ackout(X), Y) -> ACKIN(Y, X)
ACKIN(s(s(X')), s(0)) -> U21(u11(u21(ackin(s(X'), 0), X')), s(X'))
ackin(0, X) -> ackout(s(X))
ackin(s(X), 0) -> u11(ackin(X, s(0)))
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u11(ackout(X)) -> ackout(X)
u21(ackout(X), Y) -> u22(ackin(Y, X))
u22(ackout(X)) -> ackout(X)
innermost
one new Dependency Pair is created:
ACKIN(s(s(X')), s(0)) -> U21(u11(u21(ackin(s(X'), 0), X')), s(X'))
ACKIN(s(s(X')), s(0)) -> U21(u11(u21(u11(ackin(X', s(0))), X')), s(X'))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 5
↳Narrowing Transformation
ACKIN(s(s(X')), s(0)) -> U21(u11(u21(u11(ackin(X', s(0))), X')), s(X'))
ACKIN(s(X''), s(s(Y''))) -> U21(u21(ackin(s(X''), Y''), X''), X'')
ACKIN(s(X), s(Y)) -> ACKIN(s(X), Y)
ACKIN(s(X), 0) -> ACKIN(X, s(0))
U21(ackout(X), Y) -> ACKIN(Y, X)
ACKIN(s(0), s(0)) -> U21(ackout(s(s(0))), 0)
ackin(0, X) -> ackout(s(X))
ackin(s(X), 0) -> u11(ackin(X, s(0)))
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u11(ackout(X)) -> ackout(X)
u21(ackout(X), Y) -> u22(ackin(Y, X))
u22(ackout(X)) -> ackout(X)
innermost
two new Dependency Pairs are created:
ACKIN(s(X''), s(s(Y''))) -> U21(u21(ackin(s(X''), Y''), X''), X'')
ACKIN(s(X'''), s(s(0))) -> U21(u21(u11(ackin(X''', s(0))), X'''), X''')
ACKIN(s(X'''), s(s(s(Y')))) -> U21(u21(u21(ackin(s(X'''), Y'), X'''), X'''), X''')
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 6
↳Forward Instantiation Transformation
ACKIN(s(X'''), s(s(s(Y')))) -> U21(u21(u21(ackin(s(X'''), Y'), X'''), X'''), X''')
ACKIN(s(X'''), s(s(0))) -> U21(u21(u11(ackin(X''', s(0))), X'''), X''')
ACKIN(s(0), s(0)) -> U21(ackout(s(s(0))), 0)
ACKIN(s(X), s(Y)) -> ACKIN(s(X), Y)
ACKIN(s(X), 0) -> ACKIN(X, s(0))
U21(ackout(X), Y) -> ACKIN(Y, X)
ACKIN(s(s(X')), s(0)) -> U21(u11(u21(u11(ackin(X', s(0))), X')), s(X'))
ackin(0, X) -> ackout(s(X))
ackin(s(X), 0) -> u11(ackin(X, s(0)))
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u11(ackout(X)) -> ackout(X)
u21(ackout(X), Y) -> u22(ackin(Y, X))
u22(ackout(X)) -> ackout(X)
innermost
three new Dependency Pairs are created:
ACKIN(s(X), 0) -> ACKIN(X, s(0))
ACKIN(s(s(X'')), 0) -> ACKIN(s(X''), s(0))
ACKIN(s(s(0)), 0) -> ACKIN(s(0), s(0))
ACKIN(s(s(s(X'''))), 0) -> ACKIN(s(s(X''')), s(0))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 7
↳Forward Instantiation Transformation
ACKIN(s(s(s(X'''))), 0) -> ACKIN(s(s(X''')), s(0))
ACKIN(s(s(0)), 0) -> ACKIN(s(0), s(0))
ACKIN(s(s(X'')), 0) -> ACKIN(s(X''), s(0))
ACKIN(s(X'''), s(s(0))) -> U21(u21(u11(ackin(X''', s(0))), X'''), X''')
ACKIN(s(s(X')), s(0)) -> U21(u11(u21(u11(ackin(X', s(0))), X')), s(X'))
ACKIN(s(0), s(0)) -> U21(ackout(s(s(0))), 0)
ACKIN(s(X), s(Y)) -> ACKIN(s(X), Y)
U21(ackout(X), Y) -> ACKIN(Y, X)
ACKIN(s(X'''), s(s(s(Y')))) -> U21(u21(u21(ackin(s(X'''), Y'), X'''), X'''), X''')
ackin(0, X) -> ackout(s(X))
ackin(s(X), 0) -> u11(ackin(X, s(0)))
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u11(ackout(X)) -> ackout(X)
u21(ackout(X), Y) -> u22(ackin(Y, X))
u22(ackout(X)) -> ackout(X)
innermost
eight new Dependency Pairs are created:
ACKIN(s(X), s(Y)) -> ACKIN(s(X), Y)
ACKIN(s(X''), s(s(Y''))) -> ACKIN(s(X''), s(Y''))
ACKIN(s(0), s(s(0))) -> ACKIN(s(0), s(0))
ACKIN(s(s(X''')), s(s(0))) -> ACKIN(s(s(X''')), s(0))
ACKIN(s(X'), s(s(s(0)))) -> ACKIN(s(X'), s(s(0)))
ACKIN(s(X'), s(s(s(s(Y'''))))) -> ACKIN(s(X'), s(s(s(Y'''))))
ACKIN(s(s(X'''')), s(0)) -> ACKIN(s(s(X'''')), 0)
ACKIN(s(s(0)), s(0)) -> ACKIN(s(s(0)), 0)
ACKIN(s(s(s(X'''''))), s(0)) -> ACKIN(s(s(s(X'''''))), 0)
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 8
↳Forward Instantiation Transformation
ACKIN(s(X'), s(s(s(s(Y'''))))) -> ACKIN(s(X'), s(s(s(Y'''))))
ACKIN(s(X'), s(s(s(0)))) -> ACKIN(s(X'), s(s(0)))
ACKIN(s(s(X''')), s(s(0))) -> ACKIN(s(s(X''')), s(0))
ACKIN(s(0), s(s(0))) -> ACKIN(s(0), s(0))
ACKIN(s(X''), s(s(Y''))) -> ACKIN(s(X''), s(Y''))
ACKIN(s(s(s(X'''''))), s(0)) -> ACKIN(s(s(s(X'''''))), 0)
ACKIN(s(s(0)), s(0)) -> ACKIN(s(s(0)), 0)
ACKIN(s(s(0)), 0) -> ACKIN(s(0), s(0))
ACKIN(s(s(X'''')), s(0)) -> ACKIN(s(s(X'''')), 0)
ACKIN(s(s(X'')), 0) -> ACKIN(s(X''), s(0))
ACKIN(s(X'''), s(s(s(Y')))) -> U21(u21(u21(ackin(s(X'''), Y'), X'''), X'''), X''')
ACKIN(s(X'''), s(s(0))) -> U21(u21(u11(ackin(X''', s(0))), X'''), X''')
ACKIN(s(0), s(0)) -> U21(ackout(s(s(0))), 0)
U21(ackout(X), Y) -> ACKIN(Y, X)
ACKIN(s(s(X')), s(0)) -> U21(u11(u21(u11(ackin(X', s(0))), X')), s(X'))
ACKIN(s(s(s(X'''))), 0) -> ACKIN(s(s(X''')), s(0))
ackin(0, X) -> ackout(s(X))
ackin(s(X), 0) -> u11(ackin(X, s(0)))
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u11(ackout(X)) -> ackout(X)
u21(ackout(X), Y) -> u22(ackin(Y, X))
u22(ackout(X)) -> ackout(X)
innermost
15 new Dependency Pairs are created:
U21(ackout(X), Y) -> ACKIN(Y, X)
U21(ackout(s(0)), s(0)) -> ACKIN(s(0), s(0))
U21(ackout(s(0)), s(s(X'''))) -> ACKIN(s(s(X''')), s(0))
U21(ackout(s(s(0))), s(X''''')) -> ACKIN(s(X'''''), s(s(0)))
U21(ackout(s(s(s(Y''')))), s(X''''')) -> ACKIN(s(X'''''), s(s(s(Y'''))))
U21(ackout(0), s(s(X''''))) -> ACKIN(s(s(X'''')), 0)
U21(ackout(0), s(s(0))) -> ACKIN(s(s(0)), 0)
U21(ackout(0), s(s(s(X''''')))) -> ACKIN(s(s(s(X'''''))), 0)
U21(ackout(s(s(Y''''))), s(X'''')) -> ACKIN(s(X''''), s(s(Y'''')))
U21(ackout(s(s(0))), s(0)) -> ACKIN(s(0), s(s(0)))
U21(ackout(s(s(0))), s(s(X'''''))) -> ACKIN(s(s(X''''')), s(s(0)))
U21(ackout(s(s(s(0)))), s(X''')) -> ACKIN(s(X'''), s(s(s(0))))
U21(ackout(s(s(s(s(Y'''''))))), s(X''')) -> ACKIN(s(X'''), s(s(s(s(Y''''')))))
U21(ackout(s(0)), s(s(X''''''))) -> ACKIN(s(s(X'''''')), s(0))
U21(ackout(s(0)), s(s(0))) -> ACKIN(s(s(0)), s(0))
U21(ackout(s(0)), s(s(s(X''''''')))) -> ACKIN(s(s(s(X'''''''))), s(0))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 9
↳Forward Instantiation Transformation
U21(ackout(s(0)), s(s(s(X''''''')))) -> ACKIN(s(s(s(X'''''''))), s(0))
U21(ackout(s(0)), s(s(0))) -> ACKIN(s(s(0)), s(0))
U21(ackout(s(0)), s(s(X''''''))) -> ACKIN(s(s(X'''''')), s(0))
U21(ackout(s(s(s(s(Y'''''))))), s(X''')) -> ACKIN(s(X'''), s(s(s(s(Y''''')))))
U21(ackout(s(s(s(0)))), s(X''')) -> ACKIN(s(X'''), s(s(s(0))))
U21(ackout(s(s(0))), s(s(X'''''))) -> ACKIN(s(s(X''''')), s(s(0)))
U21(ackout(s(s(0))), s(0)) -> ACKIN(s(0), s(s(0)))
U21(ackout(s(s(Y''''))), s(X'''')) -> ACKIN(s(X''''), s(s(Y'''')))
U21(ackout(0), s(s(s(X''''')))) -> ACKIN(s(s(s(X'''''))), 0)
U21(ackout(0), s(s(0))) -> ACKIN(s(s(0)), 0)
U21(ackout(0), s(s(X''''))) -> ACKIN(s(s(X'''')), 0)
ACKIN(s(X'), s(s(s(0)))) -> ACKIN(s(X'), s(s(0)))
ACKIN(s(s(s(X'''))), 0) -> ACKIN(s(s(X''')), s(0))
ACKIN(s(s(s(X'''''))), s(0)) -> ACKIN(s(s(s(X'''''))), 0)
ACKIN(s(s(0)), s(0)) -> ACKIN(s(s(0)), 0)
ACKIN(s(s(X'')), 0) -> ACKIN(s(X''), s(0))
ACKIN(s(s(X'''')), s(0)) -> ACKIN(s(s(X'''')), 0)
ACKIN(s(s(X''')), s(s(0))) -> ACKIN(s(s(X''')), s(0))
ACKIN(s(X''), s(s(Y''))) -> ACKIN(s(X''), s(Y''))
U21(ackout(s(s(s(Y''')))), s(X''''')) -> ACKIN(s(X'''''), s(s(s(Y'''))))
ACKIN(s(X'''), s(s(0))) -> U21(u21(u11(ackin(X''', s(0))), X'''), X''')
U21(ackout(s(s(0))), s(X''''')) -> ACKIN(s(X'''''), s(s(0)))
ACKIN(s(s(X')), s(0)) -> U21(u11(u21(u11(ackin(X', s(0))), X')), s(X'))
U21(ackout(s(0)), s(s(X'''))) -> ACKIN(s(s(X''')), s(0))
ACKIN(s(X'''), s(s(s(Y')))) -> U21(u21(u21(ackin(s(X'''), Y'), X'''), X'''), X''')
ACKIN(s(X'), s(s(s(s(Y'''))))) -> ACKIN(s(X'), s(s(s(Y'''))))
ackin(0, X) -> ackout(s(X))
ackin(s(X), 0) -> u11(ackin(X, s(0)))
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u11(ackout(X)) -> ackout(X)
u21(ackout(X), Y) -> u22(ackin(Y, X))
u22(ackout(X)) -> ackout(X)
innermost
four new Dependency Pairs are created:
ACKIN(s(s(X'')), 0) -> ACKIN(s(X''), s(0))
ACKIN(s(s(s(X''''))), 0) -> ACKIN(s(s(X'''')), s(0))
ACKIN(s(s(s(X''''''))), 0) -> ACKIN(s(s(X'''''')), s(0))
ACKIN(s(s(s(0))), 0) -> ACKIN(s(s(0)), s(0))
ACKIN(s(s(s(s(X''''''')))), 0) -> ACKIN(s(s(s(X'''''''))), s(0))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 10
↳Forward Instantiation Transformation
U21(ackout(s(0)), s(s(0))) -> ACKIN(s(s(0)), s(0))
U21(ackout(s(0)), s(s(X''''''))) -> ACKIN(s(s(X'''''')), s(0))
U21(ackout(s(s(s(s(Y'''''))))), s(X''')) -> ACKIN(s(X'''), s(s(s(s(Y''''')))))
U21(ackout(s(s(s(0)))), s(X''')) -> ACKIN(s(X'''), s(s(s(0))))
U21(ackout(s(s(0))), s(s(X'''''))) -> ACKIN(s(s(X''''')), s(s(0)))
U21(ackout(s(s(0))), s(0)) -> ACKIN(s(0), s(s(0)))
ACKIN(s(X'), s(s(s(s(Y'''))))) -> ACKIN(s(X'), s(s(s(Y'''))))
ACKIN(s(X'), s(s(s(0)))) -> ACKIN(s(X'), s(s(0)))
ACKIN(s(s(X''')), s(s(0))) -> ACKIN(s(s(X''')), s(0))
ACKIN(s(X''), s(s(Y''))) -> ACKIN(s(X''), s(Y''))
U21(ackout(s(s(Y''''))), s(X'''')) -> ACKIN(s(X''''), s(s(Y'''')))
U21(ackout(0), s(s(s(X''''')))) -> ACKIN(s(s(s(X'''''))), 0)
U21(ackout(0), s(s(X''''))) -> ACKIN(s(s(X'''')), 0)
ACKIN(s(X'''), s(s(s(Y')))) -> U21(u21(u21(ackin(s(X'''), Y'), X'''), X'''), X''')
U21(ackout(s(s(s(Y''')))), s(X''''')) -> ACKIN(s(X'''''), s(s(s(Y'''))))
ACKIN(s(X'''), s(s(0))) -> U21(u21(u11(ackin(X''', s(0))), X'''), X''')
U21(ackout(s(s(0))), s(X''''')) -> ACKIN(s(X'''''), s(s(0)))
ACKIN(s(s(s(s(X''''''')))), 0) -> ACKIN(s(s(s(X'''''''))), s(0))
ACKIN(s(s(s(0))), 0) -> ACKIN(s(s(0)), s(0))
ACKIN(s(s(s(X''''''))), 0) -> ACKIN(s(s(X'''''')), s(0))
ACKIN(s(s(s(X''''))), 0) -> ACKIN(s(s(X'''')), s(0))
ACKIN(s(s(s(X'''''))), s(0)) -> ACKIN(s(s(s(X'''''))), 0)
ACKIN(s(s(s(X'''))), 0) -> ACKIN(s(s(X''')), s(0))
ACKIN(s(s(X'''')), s(0)) -> ACKIN(s(s(X'''')), 0)
U21(ackout(s(0)), s(s(X'''))) -> ACKIN(s(s(X''')), s(0))
ACKIN(s(s(X')), s(0)) -> U21(u11(u21(u11(ackin(X', s(0))), X')), s(X'))
U21(ackout(s(0)), s(s(s(X''''''')))) -> ACKIN(s(s(s(X'''''''))), s(0))
ackin(0, X) -> ackout(s(X))
ackin(s(X), 0) -> u11(ackin(X, s(0)))
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u11(ackout(X)) -> ackout(X)
u21(ackout(X), Y) -> u22(ackin(Y, X))
u22(ackout(X)) -> ackout(X)
innermost
nine new Dependency Pairs are created:
ACKIN(s(X''), s(s(Y''))) -> ACKIN(s(X''), s(Y''))
ACKIN(s(s(X'''')), s(s(0))) -> ACKIN(s(s(X'''')), s(0))
ACKIN(s(X'''), s(s(s(0)))) -> ACKIN(s(X'''), s(s(0)))
ACKIN(s(X'''), s(s(s(s(Y''''))))) -> ACKIN(s(X'''), s(s(s(Y''''))))
ACKIN(s(X''''), s(s(s(Y'''')))) -> ACKIN(s(X''''), s(s(Y'''')))
ACKIN(s(s(X''''')), s(s(s(0)))) -> ACKIN(s(s(X''''')), s(s(0)))
ACKIN(s(X''''), s(s(s(s(0))))) -> ACKIN(s(X''''), s(s(s(0))))
ACKIN(s(X''''), s(s(s(s(s(Y''''')))))) -> ACKIN(s(X''''), s(s(s(s(Y''''')))))
ACKIN(s(s(X'''''')), s(s(0))) -> ACKIN(s(s(X'''''')), s(0))
ACKIN(s(s(s(X'''''''))), s(s(0))) -> ACKIN(s(s(s(X'''''''))), s(0))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 11
↳Forward Instantiation Transformation
U21(ackout(s(0)), s(s(s(X''''''')))) -> ACKIN(s(s(s(X'''''''))), s(0))
U21(ackout(s(0)), s(s(X''''''))) -> ACKIN(s(s(X'''''')), s(0))
U21(ackout(s(s(s(s(Y'''''))))), s(X''')) -> ACKIN(s(X'''), s(s(s(s(Y''''')))))
U21(ackout(s(s(s(0)))), s(X''')) -> ACKIN(s(X'''), s(s(s(0))))
U21(ackout(s(s(0))), s(s(X'''''))) -> ACKIN(s(s(X''''')), s(s(0)))
U21(ackout(s(s(0))), s(0)) -> ACKIN(s(0), s(s(0)))
ACKIN(s(X''''), s(s(s(s(s(Y''''')))))) -> ACKIN(s(X''''), s(s(s(s(Y''''')))))
ACKIN(s(X''''), s(s(s(s(0))))) -> ACKIN(s(X''''), s(s(s(0))))
ACKIN(s(s(X''''')), s(s(s(0)))) -> ACKIN(s(s(X''''')), s(s(0)))
ACKIN(s(X''''), s(s(s(Y'''')))) -> ACKIN(s(X''''), s(s(Y'''')))
ACKIN(s(X'''), s(s(s(s(Y''''))))) -> ACKIN(s(X'''), s(s(s(Y''''))))
ACKIN(s(X'''), s(s(s(0)))) -> ACKIN(s(X'''), s(s(0)))
ACKIN(s(X'), s(s(s(s(Y'''))))) -> ACKIN(s(X'), s(s(s(Y'''))))
ACKIN(s(s(s(X'''''''))), s(s(0))) -> ACKIN(s(s(s(X'''''''))), s(0))
ACKIN(s(s(X'''''')), s(s(0))) -> ACKIN(s(s(X'''''')), s(0))
ACKIN(s(s(X'''')), s(s(0))) -> ACKIN(s(s(X'''')), s(0))
ACKIN(s(X'), s(s(s(0)))) -> ACKIN(s(X'), s(s(0)))
ACKIN(s(s(X''')), s(s(0))) -> ACKIN(s(s(X''')), s(0))
U21(ackout(s(s(Y''''))), s(X'''')) -> ACKIN(s(X''''), s(s(Y'''')))
U21(ackout(0), s(s(s(X''''')))) -> ACKIN(s(s(s(X'''''))), 0)
U21(ackout(0), s(s(X''''))) -> ACKIN(s(s(X'''')), 0)
ACKIN(s(X'''), s(s(s(Y')))) -> U21(u21(u21(ackin(s(X'''), Y'), X'''), X'''), X''')
U21(ackout(s(s(s(Y''')))), s(X''''')) -> ACKIN(s(X'''''), s(s(s(Y'''))))
ACKIN(s(X'''), s(s(0))) -> U21(u21(u11(ackin(X''', s(0))), X'''), X''')
U21(ackout(s(s(0))), s(X''''')) -> ACKIN(s(X'''''), s(s(0)))
ACKIN(s(s(s(s(X''''''')))), 0) -> ACKIN(s(s(s(X'''''''))), s(0))
ACKIN(s(s(s(0))), 0) -> ACKIN(s(s(0)), s(0))
ACKIN(s(s(s(X''''''))), 0) -> ACKIN(s(s(X'''''')), s(0))
ACKIN(s(s(s(X''''))), 0) -> ACKIN(s(s(X'''')), s(0))
ACKIN(s(s(s(X'''''))), s(0)) -> ACKIN(s(s(s(X'''''))), 0)
ACKIN(s(s(s(X'''))), 0) -> ACKIN(s(s(X''')), s(0))
ACKIN(s(s(X'''')), s(0)) -> ACKIN(s(s(X'''')), 0)
U21(ackout(s(0)), s(s(X'''))) -> ACKIN(s(s(X''')), s(0))
ACKIN(s(s(X')), s(0)) -> U21(u11(u21(u11(ackin(X', s(0))), X')), s(X'))
U21(ackout(s(0)), s(s(0))) -> ACKIN(s(s(0)), s(0))
ackin(0, X) -> ackout(s(X))
ackin(s(X), 0) -> u11(ackin(X, s(0)))
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u11(ackout(X)) -> ackout(X)
u21(ackout(X), Y) -> u22(ackin(Y, X))
u22(ackout(X)) -> ackout(X)
innermost
four new Dependency Pairs are created:
ACKIN(s(s(X'''')), s(0)) -> ACKIN(s(s(X'''')), 0)
ACKIN(s(s(s(X''''''))), s(0)) -> ACKIN(s(s(s(X''''''))), 0)
ACKIN(s(s(s(X''''''''))), s(0)) -> ACKIN(s(s(s(X''''''''))), 0)
ACKIN(s(s(s(0))), s(0)) -> ACKIN(s(s(s(0))), 0)
ACKIN(s(s(s(s(X''''''''')))), s(0)) -> ACKIN(s(s(s(s(X''''''''')))), 0)
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 12
↳Forward Instantiation Transformation
U21(ackout(s(0)), s(s(0))) -> ACKIN(s(s(0)), s(0))
U21(ackout(s(0)), s(s(X''''''))) -> ACKIN(s(s(X'''''')), s(0))
U21(ackout(s(s(s(s(Y'''''))))), s(X''')) -> ACKIN(s(X'''), s(s(s(s(Y''''')))))
U21(ackout(s(s(s(0)))), s(X''')) -> ACKIN(s(X'''), s(s(s(0))))
U21(ackout(s(s(0))), s(s(X'''''))) -> ACKIN(s(s(X''''')), s(s(0)))
U21(ackout(s(s(0))), s(0)) -> ACKIN(s(0), s(s(0)))
ACKIN(s(X''''), s(s(s(s(s(Y''''')))))) -> ACKIN(s(X''''), s(s(s(s(Y''''')))))
ACKIN(s(X''''), s(s(s(s(0))))) -> ACKIN(s(X''''), s(s(s(0))))
ACKIN(s(s(X''''')), s(s(s(0)))) -> ACKIN(s(s(X''''')), s(s(0)))
ACKIN(s(X''''), s(s(s(Y'''')))) -> ACKIN(s(X''''), s(s(Y'''')))
ACKIN(s(X'''), s(s(s(s(Y''''))))) -> ACKIN(s(X'''), s(s(s(Y''''))))
ACKIN(s(X'''), s(s(s(0)))) -> ACKIN(s(X'''), s(s(0)))
ACKIN(s(X'), s(s(s(s(Y'''))))) -> ACKIN(s(X'), s(s(s(Y'''))))
ACKIN(s(s(s(X'''''''))), s(s(0))) -> ACKIN(s(s(s(X'''''''))), s(0))
ACKIN(s(s(X'''''')), s(s(0))) -> ACKIN(s(s(X'''''')), s(0))
ACKIN(s(s(X'''')), s(s(0))) -> ACKIN(s(s(X'''')), s(0))
ACKIN(s(X'), s(s(s(0)))) -> ACKIN(s(X'), s(s(0)))
ACKIN(s(s(X''')), s(s(0))) -> ACKIN(s(s(X''')), s(0))
U21(ackout(s(s(Y''''))), s(X'''')) -> ACKIN(s(X''''), s(s(Y'''')))
U21(ackout(0), s(s(s(X''''')))) -> ACKIN(s(s(s(X'''''))), 0)
U21(ackout(0), s(s(X''''))) -> ACKIN(s(s(X'''')), 0)
ACKIN(s(X'''), s(s(s(Y')))) -> U21(u21(u21(ackin(s(X'''), Y'), X'''), X'''), X''')
U21(ackout(s(s(s(Y''')))), s(X''''')) -> ACKIN(s(X'''''), s(s(s(Y'''))))
ACKIN(s(X'''), s(s(0))) -> U21(u21(u11(ackin(X''', s(0))), X'''), X''')
U21(ackout(s(s(0))), s(X''''')) -> ACKIN(s(X'''''), s(s(0)))
ACKIN(s(s(s(s(X''''''')))), 0) -> ACKIN(s(s(s(X'''''''))), s(0))
ACKIN(s(s(s(s(X''''''''')))), s(0)) -> ACKIN(s(s(s(s(X''''''''')))), 0)
ACKIN(s(s(s(0))), 0) -> ACKIN(s(s(0)), s(0))
ACKIN(s(s(s(0))), s(0)) -> ACKIN(s(s(s(0))), 0)
ACKIN(s(s(s(X''''''))), 0) -> ACKIN(s(s(X'''''')), s(0))
ACKIN(s(s(s(X''''''''))), s(0)) -> ACKIN(s(s(s(X''''''''))), 0)
ACKIN(s(s(s(X''''))), 0) -> ACKIN(s(s(X'''')), s(0))
ACKIN(s(s(s(X''''''))), s(0)) -> ACKIN(s(s(s(X''''''))), 0)
ACKIN(s(s(s(X'''))), 0) -> ACKIN(s(s(X''')), s(0))
ACKIN(s(s(s(X'''''))), s(0)) -> ACKIN(s(s(s(X'''''))), 0)
U21(ackout(s(0)), s(s(X'''))) -> ACKIN(s(s(X''')), s(0))
ACKIN(s(s(X')), s(0)) -> U21(u11(u21(u11(ackin(X', s(0))), X')), s(X'))
U21(ackout(s(0)), s(s(s(X''''''')))) -> ACKIN(s(s(s(X'''''''))), s(0))
ackin(0, X) -> ackout(s(X))
ackin(s(X), 0) -> u11(ackin(X, s(0)))
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u11(ackout(X)) -> ackout(X)
u21(ackout(X), Y) -> u22(ackin(Y, X))
u22(ackout(X)) -> ackout(X)
innermost
four new Dependency Pairs are created:
U21(ackout(0), s(s(X''''))) -> ACKIN(s(s(X'''')), 0)
U21(ackout(0), s(s(s(X'''''')))) -> ACKIN(s(s(s(X''''''))), 0)
U21(ackout(0), s(s(s(X'''''''')))) -> ACKIN(s(s(s(X''''''''))), 0)
U21(ackout(0), s(s(s(0)))) -> ACKIN(s(s(s(0))), 0)
U21(ackout(0), s(s(s(s(X'''''''''))))) -> ACKIN(s(s(s(s(X''''''''')))), 0)
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 13
↳Forward Instantiation Transformation
U21(ackout(0), s(s(s(s(X'''''''''))))) -> ACKIN(s(s(s(s(X''''''''')))), 0)
U21(ackout(0), s(s(s(0)))) -> ACKIN(s(s(s(0))), 0)
U21(ackout(0), s(s(s(X'''''''')))) -> ACKIN(s(s(s(X''''''''))), 0)
U21(ackout(0), s(s(s(X'''''')))) -> ACKIN(s(s(s(X''''''))), 0)
U21(ackout(s(0)), s(s(s(X''''''')))) -> ACKIN(s(s(s(X'''''''))), s(0))
U21(ackout(s(0)), s(s(X''''''))) -> ACKIN(s(s(X'''''')), s(0))
U21(ackout(s(s(s(s(Y'''''))))), s(X''')) -> ACKIN(s(X'''), s(s(s(s(Y''''')))))
U21(ackout(s(s(s(0)))), s(X''')) -> ACKIN(s(X'''), s(s(s(0))))
U21(ackout(s(s(0))), s(s(X'''''))) -> ACKIN(s(s(X''''')), s(s(0)))
U21(ackout(s(s(0))), s(0)) -> ACKIN(s(0), s(s(0)))
ACKIN(s(X''''), s(s(s(s(s(Y''''')))))) -> ACKIN(s(X''''), s(s(s(s(Y''''')))))
ACKIN(s(X''''), s(s(s(s(0))))) -> ACKIN(s(X''''), s(s(s(0))))
ACKIN(s(s(X''''')), s(s(s(0)))) -> ACKIN(s(s(X''''')), s(s(0)))
ACKIN(s(X''''), s(s(s(Y'''')))) -> ACKIN(s(X''''), s(s(Y'''')))
ACKIN(s(X'''), s(s(s(s(Y''''))))) -> ACKIN(s(X'''), s(s(s(Y''''))))
ACKIN(s(X'''), s(s(s(0)))) -> ACKIN(s(X'''), s(s(0)))
ACKIN(s(X'), s(s(s(s(Y'''))))) -> ACKIN(s(X'), s(s(s(Y'''))))
ACKIN(s(s(s(X'''''''))), s(s(0))) -> ACKIN(s(s(s(X'''''''))), s(0))
ACKIN(s(s(X'''''')), s(s(0))) -> ACKIN(s(s(X'''''')), s(0))
ACKIN(s(s(X'''')), s(s(0))) -> ACKIN(s(s(X'''')), s(0))
ACKIN(s(X'), s(s(s(0)))) -> ACKIN(s(X'), s(s(0)))
ACKIN(s(s(X''')), s(s(0))) -> ACKIN(s(s(X''')), s(0))
U21(ackout(s(s(Y''''))), s(X'''')) -> ACKIN(s(X''''), s(s(Y'''')))
U21(ackout(0), s(s(s(X''''')))) -> ACKIN(s(s(s(X'''''))), 0)
ACKIN(s(X'''), s(s(s(Y')))) -> U21(u21(u21(ackin(s(X'''), Y'), X'''), X'''), X''')
U21(ackout(s(s(s(Y''')))), s(X''''')) -> ACKIN(s(X'''''), s(s(s(Y'''))))
ACKIN(s(X'''), s(s(0))) -> U21(u21(u11(ackin(X''', s(0))), X'''), X''')
U21(ackout(s(s(0))), s(X''''')) -> ACKIN(s(X'''''), s(s(0)))
ACKIN(s(s(s(s(X''''''')))), 0) -> ACKIN(s(s(s(X'''''''))), s(0))
ACKIN(s(s(s(s(X''''''''')))), s(0)) -> ACKIN(s(s(s(s(X''''''''')))), 0)
ACKIN(s(s(s(0))), 0) -> ACKIN(s(s(0)), s(0))
ACKIN(s(s(s(0))), s(0)) -> ACKIN(s(s(s(0))), 0)
ACKIN(s(s(s(X''''''))), 0) -> ACKIN(s(s(X'''''')), s(0))
ACKIN(s(s(s(X''''''''))), s(0)) -> ACKIN(s(s(s(X''''''''))), 0)
ACKIN(s(s(s(X''''))), 0) -> ACKIN(s(s(X'''')), s(0))
ACKIN(s(s(s(X''''''))), s(0)) -> ACKIN(s(s(s(X''''''))), 0)
ACKIN(s(s(s(X'''))), 0) -> ACKIN(s(s(X''')), s(0))
ACKIN(s(s(s(X'''''))), s(0)) -> ACKIN(s(s(s(X'''''))), 0)
U21(ackout(s(0)), s(s(X'''))) -> ACKIN(s(s(X''')), s(0))
ACKIN(s(s(X')), s(0)) -> U21(u11(u21(u11(ackin(X', s(0))), X')), s(X'))
U21(ackout(s(0)), s(s(0))) -> ACKIN(s(s(0)), s(0))
ackin(0, X) -> ackout(s(X))
ackin(s(X), 0) -> u11(ackin(X, s(0)))
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u11(ackout(X)) -> ackout(X)
u21(ackout(X), Y) -> u22(ackin(Y, X))
u22(ackout(X)) -> ackout(X)
innermost
13 new Dependency Pairs are created:
U21(ackout(s(s(Y''''))), s(X'''')) -> ACKIN(s(X''''), s(s(Y'''')))
U21(ackout(s(s(0))), s(X'''''')) -> ACKIN(s(X''''''), s(s(0)))
U21(ackout(s(s(s(Y''')))), s(X'''''')) -> ACKIN(s(X''''''), s(s(s(Y'''))))
U21(ackout(s(s(0))), s(s(X''''''))) -> ACKIN(s(s(X'''''')), s(s(0)))
U21(ackout(s(s(s(0)))), s(X''''')) -> ACKIN(s(X'''''), s(s(s(0))))
U21(ackout(s(s(s(s(Y''''''))))), s(X''''')) -> ACKIN(s(X'''''), s(s(s(s(Y'''''')))))
U21(ackout(s(s(s(0)))), s(X'''''')) -> ACKIN(s(X''''''), s(s(s(0))))
U21(ackout(s(s(s(s(Y''''''))))), s(X'''''')) -> ACKIN(s(X''''''), s(s(s(s(Y'''''')))))
U21(ackout(s(s(s(Y'''''')))), s(X'''''')) -> ACKIN(s(X''''''), s(s(s(Y''''''))))
U21(ackout(s(s(s(0)))), s(s(X'''''''))) -> ACKIN(s(s(X''''''')), s(s(s(0))))
U21(ackout(s(s(s(s(0))))), s(X'''''')) -> ACKIN(s(X''''''), s(s(s(s(0)))))
U21(ackout(s(s(s(s(s(Y''''''')))))), s(X'''''')) -> ACKIN(s(X''''''), s(s(s(s(s(Y'''''''))))))
U21(ackout(s(s(0))), s(s(X''''''''))) -> ACKIN(s(s(X'''''''')), s(s(0)))
U21(ackout(s(s(0))), s(s(s(X''''''''')))) -> ACKIN(s(s(s(X'''''''''))), s(s(0)))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 14
↳Polynomial Ordering
U21(ackout(s(s(0))), s(s(s(X''''''''')))) -> ACKIN(s(s(s(X'''''''''))), s(s(0)))
U21(ackout(s(s(0))), s(s(X''''''''))) -> ACKIN(s(s(X'''''''')), s(s(0)))
U21(ackout(s(s(s(s(s(Y''''''')))))), s(X'''''')) -> ACKIN(s(X''''''), s(s(s(s(s(Y'''''''))))))
U21(ackout(s(s(s(s(0))))), s(X'''''')) -> ACKIN(s(X''''''), s(s(s(s(0)))))
U21(ackout(s(s(s(0)))), s(s(X'''''''))) -> ACKIN(s(s(X''''''')), s(s(s(0))))
U21(ackout(s(s(s(Y'''''')))), s(X'''''')) -> ACKIN(s(X''''''), s(s(s(Y''''''))))
U21(ackout(s(s(s(s(Y''''''))))), s(X'''''')) -> ACKIN(s(X''''''), s(s(s(s(Y'''''')))))
U21(ackout(s(s(s(0)))), s(X'''''')) -> ACKIN(s(X''''''), s(s(s(0))))
U21(ackout(s(s(s(s(Y''''''))))), s(X''''')) -> ACKIN(s(X'''''), s(s(s(s(Y'''''')))))
U21(ackout(s(s(s(0)))), s(X''''')) -> ACKIN(s(X'''''), s(s(s(0))))
U21(ackout(s(s(0))), s(s(X''''''))) -> ACKIN(s(s(X'''''')), s(s(0)))
U21(ackout(s(s(s(Y''')))), s(X'''''')) -> ACKIN(s(X''''''), s(s(s(Y'''))))
U21(ackout(s(s(0))), s(X'''''')) -> ACKIN(s(X''''''), s(s(0)))
U21(ackout(0), s(s(s(0)))) -> ACKIN(s(s(s(0))), 0)
U21(ackout(0), s(s(s(X'''''''')))) -> ACKIN(s(s(s(X''''''''))), 0)
U21(ackout(0), s(s(s(X'''''')))) -> ACKIN(s(s(s(X''''''))), 0)
U21(ackout(s(0)), s(s(s(X''''''')))) -> ACKIN(s(s(s(X'''''''))), s(0))
U21(ackout(s(0)), s(s(0))) -> ACKIN(s(s(0)), s(0))
U21(ackout(s(0)), s(s(X''''''))) -> ACKIN(s(s(X'''''')), s(0))
U21(ackout(s(s(s(s(Y'''''))))), s(X''')) -> ACKIN(s(X'''), s(s(s(s(Y''''')))))
ACKIN(s(X''''), s(s(s(s(s(Y''''')))))) -> ACKIN(s(X''''), s(s(s(s(Y''''')))))
ACKIN(s(X''''), s(s(s(s(0))))) -> ACKIN(s(X''''), s(s(s(0))))
ACKIN(s(s(X''''')), s(s(s(0)))) -> ACKIN(s(s(X''''')), s(s(0)))
ACKIN(s(X'''), s(s(s(s(Y''''))))) -> ACKIN(s(X'''), s(s(s(Y''''))))
ACKIN(s(X'), s(s(s(s(Y'''))))) -> ACKIN(s(X'), s(s(s(Y'''))))
ACKIN(s(X''''), s(s(s(Y'''')))) -> ACKIN(s(X''''), s(s(Y'''')))
ACKIN(s(X'''), s(s(s(0)))) -> ACKIN(s(X'''), s(s(0)))
ACKIN(s(X'), s(s(s(0)))) -> ACKIN(s(X'), s(s(0)))
U21(ackout(s(s(s(0)))), s(X''')) -> ACKIN(s(X'''), s(s(s(0))))
ACKIN(s(s(s(X'''''''))), s(s(0))) -> ACKIN(s(s(s(X'''''''))), s(0))
ACKIN(s(s(X'''''')), s(s(0))) -> ACKIN(s(s(X'''''')), s(0))
ACKIN(s(s(X'''')), s(s(0))) -> ACKIN(s(s(X'''')), s(0))
ACKIN(s(s(X''')), s(s(0))) -> ACKIN(s(s(X''')), s(0))
U21(ackout(s(s(0))), s(s(X'''''))) -> ACKIN(s(s(X''''')), s(s(0)))
U21(ackout(s(s(0))), s(0)) -> ACKIN(s(0), s(s(0)))
U21(ackout(0), s(s(s(X''''')))) -> ACKIN(s(s(s(X'''''))), 0)
ACKIN(s(X'''), s(s(s(Y')))) -> U21(u21(u21(ackin(s(X'''), Y'), X'''), X'''), X''')
U21(ackout(s(s(s(Y''')))), s(X''''')) -> ACKIN(s(X'''''), s(s(s(Y'''))))
ACKIN(s(X'''), s(s(0))) -> U21(u21(u11(ackin(X''', s(0))), X'''), X''')
U21(ackout(s(s(0))), s(X''''')) -> ACKIN(s(X'''''), s(s(0)))
ACKIN(s(s(s(s(X''''''''')))), s(0)) -> ACKIN(s(s(s(s(X''''''''')))), 0)
ACKIN(s(s(s(0))), s(0)) -> ACKIN(s(s(s(0))), 0)
ACKIN(s(s(s(s(X''''''')))), 0) -> ACKIN(s(s(s(X'''''''))), s(0))
ACKIN(s(s(s(0))), 0) -> ACKIN(s(s(0)), s(0))
ACKIN(s(s(s(X''''''''))), s(0)) -> ACKIN(s(s(s(X''''''''))), 0)
ACKIN(s(s(s(X''''''))), 0) -> ACKIN(s(s(X'''''')), s(0))
ACKIN(s(s(s(X''''''))), s(0)) -> ACKIN(s(s(s(X''''''))), 0)
ACKIN(s(s(s(X''''))), 0) -> ACKIN(s(s(X'''')), s(0))
ACKIN(s(s(s(X'''''))), s(0)) -> ACKIN(s(s(s(X'''''))), 0)
U21(ackout(s(0)), s(s(X'''))) -> ACKIN(s(s(X''')), s(0))
ACKIN(s(s(X')), s(0)) -> U21(u11(u21(u11(ackin(X', s(0))), X')), s(X'))
ACKIN(s(s(s(X'''))), 0) -> ACKIN(s(s(X''')), s(0))
U21(ackout(0), s(s(s(s(X'''''''''))))) -> ACKIN(s(s(s(s(X''''''''')))), 0)
ackin(0, X) -> ackout(s(X))
ackin(s(X), 0) -> u11(ackin(X, s(0)))
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u11(ackout(X)) -> ackout(X)
u21(ackout(X), Y) -> u22(ackin(Y, X))
u22(ackout(X)) -> ackout(X)
innermost
ACKIN(s(X'''), s(s(s(Y')))) -> U21(u21(u21(ackin(s(X'''), Y'), X'''), X'''), X''')
ACKIN(s(X'''), s(s(0))) -> U21(u21(u11(ackin(X''', s(0))), X'''), X''')
ACKIN(s(s(s(s(X''''''')))), 0) -> ACKIN(s(s(s(X'''''''))), s(0))
ACKIN(s(s(s(0))), 0) -> ACKIN(s(s(0)), s(0))
ACKIN(s(s(s(X''''''))), 0) -> ACKIN(s(s(X'''''')), s(0))
ACKIN(s(s(s(X''''))), 0) -> ACKIN(s(s(X'''')), s(0))
ACKIN(s(s(X')), s(0)) -> U21(u11(u21(u11(ackin(X', s(0))), X')), s(X'))
ACKIN(s(s(s(X'''))), 0) -> ACKIN(s(s(X''')), s(0))
POL(0) = 0 POL(u11(x1)) = 0 POL(u22(x1)) = 0 POL(U21(x1, x2)) = x2 POL(ackin(x1, x2)) = 0 POL(u21(x1, x2)) = 0 POL(s(x1)) = 1 + x1 POL(ACKIN(x1, x2)) = x1 POL(ackout(x1)) = 0
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 15
↳Dependency Graph
U21(ackout(s(s(0))), s(s(s(X''''''''')))) -> ACKIN(s(s(s(X'''''''''))), s(s(0)))
U21(ackout(s(s(0))), s(s(X''''''''))) -> ACKIN(s(s(X'''''''')), s(s(0)))
U21(ackout(s(s(s(s(s(Y''''''')))))), s(X'''''')) -> ACKIN(s(X''''''), s(s(s(s(s(Y'''''''))))))
U21(ackout(s(s(s(s(0))))), s(X'''''')) -> ACKIN(s(X''''''), s(s(s(s(0)))))
U21(ackout(s(s(s(0)))), s(s(X'''''''))) -> ACKIN(s(s(X''''''')), s(s(s(0))))
U21(ackout(s(s(s(Y'''''')))), s(X'''''')) -> ACKIN(s(X''''''), s(s(s(Y''''''))))
U21(ackout(s(s(s(s(Y''''''))))), s(X'''''')) -> ACKIN(s(X''''''), s(s(s(s(Y'''''')))))
U21(ackout(s(s(s(0)))), s(X'''''')) -> ACKIN(s(X''''''), s(s(s(0))))
U21(ackout(s(s(s(s(Y''''''))))), s(X''''')) -> ACKIN(s(X'''''), s(s(s(s(Y'''''')))))
U21(ackout(s(s(s(0)))), s(X''''')) -> ACKIN(s(X'''''), s(s(s(0))))
U21(ackout(s(s(0))), s(s(X''''''))) -> ACKIN(s(s(X'''''')), s(s(0)))
U21(ackout(s(s(s(Y''')))), s(X'''''')) -> ACKIN(s(X''''''), s(s(s(Y'''))))
U21(ackout(s(s(0))), s(X'''''')) -> ACKIN(s(X''''''), s(s(0)))
U21(ackout(0), s(s(s(0)))) -> ACKIN(s(s(s(0))), 0)
U21(ackout(0), s(s(s(X'''''''')))) -> ACKIN(s(s(s(X''''''''))), 0)
U21(ackout(0), s(s(s(X'''''')))) -> ACKIN(s(s(s(X''''''))), 0)
U21(ackout(s(0)), s(s(s(X''''''')))) -> ACKIN(s(s(s(X'''''''))), s(0))
U21(ackout(s(0)), s(s(0))) -> ACKIN(s(s(0)), s(0))
U21(ackout(s(0)), s(s(X''''''))) -> ACKIN(s(s(X'''''')), s(0))
U21(ackout(s(s(s(s(Y'''''))))), s(X''')) -> ACKIN(s(X'''), s(s(s(s(Y''''')))))
ACKIN(s(X''''), s(s(s(s(s(Y''''')))))) -> ACKIN(s(X''''), s(s(s(s(Y''''')))))
ACKIN(s(X''''), s(s(s(s(0))))) -> ACKIN(s(X''''), s(s(s(0))))
ACKIN(s(s(X''''')), s(s(s(0)))) -> ACKIN(s(s(X''''')), s(s(0)))
ACKIN(s(X'''), s(s(s(s(Y''''))))) -> ACKIN(s(X'''), s(s(s(Y''''))))
ACKIN(s(X'), s(s(s(s(Y'''))))) -> ACKIN(s(X'), s(s(s(Y'''))))
ACKIN(s(X''''), s(s(s(Y'''')))) -> ACKIN(s(X''''), s(s(Y'''')))
ACKIN(s(X'''), s(s(s(0)))) -> ACKIN(s(X'''), s(s(0)))
ACKIN(s(X'), s(s(s(0)))) -> ACKIN(s(X'), s(s(0)))
U21(ackout(s(s(s(0)))), s(X''')) -> ACKIN(s(X'''), s(s(s(0))))
ACKIN(s(s(s(X'''''''))), s(s(0))) -> ACKIN(s(s(s(X'''''''))), s(0))
ACKIN(s(s(X'''''')), s(s(0))) -> ACKIN(s(s(X'''''')), s(0))
ACKIN(s(s(X'''')), s(s(0))) -> ACKIN(s(s(X'''')), s(0))
ACKIN(s(s(X''')), s(s(0))) -> ACKIN(s(s(X''')), s(0))
U21(ackout(s(s(0))), s(s(X'''''))) -> ACKIN(s(s(X''''')), s(s(0)))
U21(ackout(s(s(0))), s(0)) -> ACKIN(s(0), s(s(0)))
U21(ackout(0), s(s(s(X''''')))) -> ACKIN(s(s(s(X'''''))), 0)
U21(ackout(s(s(s(Y''')))), s(X''''')) -> ACKIN(s(X'''''), s(s(s(Y'''))))
U21(ackout(s(s(0))), s(X''''')) -> ACKIN(s(X'''''), s(s(0)))
ACKIN(s(s(s(s(X''''''''')))), s(0)) -> ACKIN(s(s(s(s(X''''''''')))), 0)
ACKIN(s(s(s(0))), s(0)) -> ACKIN(s(s(s(0))), 0)
ACKIN(s(s(s(X''''''''))), s(0)) -> ACKIN(s(s(s(X''''''''))), 0)
ACKIN(s(s(s(X''''''))), s(0)) -> ACKIN(s(s(s(X''''''))), 0)
ACKIN(s(s(s(X'''''))), s(0)) -> ACKIN(s(s(s(X'''''))), 0)
U21(ackout(s(0)), s(s(X'''))) -> ACKIN(s(s(X''')), s(0))
U21(ackout(0), s(s(s(s(X'''''''''))))) -> ACKIN(s(s(s(s(X''''''''')))), 0)
ackin(0, X) -> ackout(s(X))
ackin(s(X), 0) -> u11(ackin(X, s(0)))
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u11(ackout(X)) -> ackout(X)
u21(ackout(X), Y) -> u22(ackin(Y, X))
u22(ackout(X)) -> ackout(X)
innermost
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 16
↳Polynomial Ordering
ACKIN(s(X''''), s(s(s(s(s(Y''''')))))) -> ACKIN(s(X''''), s(s(s(s(Y''''')))))
ACKIN(s(X''''), s(s(s(s(0))))) -> ACKIN(s(X''''), s(s(s(0))))
ACKIN(s(X''''), s(s(s(Y'''')))) -> ACKIN(s(X''''), s(s(Y'''')))
ACKIN(s(X'''), s(s(s(s(Y''''))))) -> ACKIN(s(X'''), s(s(s(Y''''))))
ACKIN(s(X'), s(s(s(s(Y'''))))) -> ACKIN(s(X'), s(s(s(Y'''))))
ackin(0, X) -> ackout(s(X))
ackin(s(X), 0) -> u11(ackin(X, s(0)))
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u11(ackout(X)) -> ackout(X)
u21(ackout(X), Y) -> u22(ackin(Y, X))
u22(ackout(X)) -> ackout(X)
innermost
ACKIN(s(X''''), s(s(s(s(s(Y''''')))))) -> ACKIN(s(X''''), s(s(s(s(Y''''')))))
ACKIN(s(X''''), s(s(s(s(0))))) -> ACKIN(s(X''''), s(s(s(0))))
ACKIN(s(X''''), s(s(s(Y'''')))) -> ACKIN(s(X''''), s(s(Y'''')))
ACKIN(s(X'''), s(s(s(s(Y''''))))) -> ACKIN(s(X'''), s(s(s(Y''''))))
ACKIN(s(X'), s(s(s(s(Y'''))))) -> ACKIN(s(X'), s(s(s(Y'''))))
POL(0) = 0 POL(s(x1)) = 1 + x1 POL(ACKIN(x1, x2)) = 1 + x1 + x2
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 17
↳Dependency Graph
ackin(0, X) -> ackout(s(X))
ackin(s(X), 0) -> u11(ackin(X, s(0)))
ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
u11(ackout(X)) -> ackout(X)
u21(ackout(X), Y) -> u22(ackin(Y, X))
u22(ackout(X)) -> ackout(X)
innermost