NO
Problem:
a(x1) -> b(c(x1))
a(b(b(x1))) -> c(a(a(a(x1))))
c(c(x1)) -> x1
Proof:
DP Processor:
DPs:
a#(x1) -> c#(x1)
a#(b(b(x1))) -> a#(x1)
a#(b(b(x1))) -> a#(a(x1))
a#(b(b(x1))) -> a#(a(a(x1)))
a#(b(b(x1))) -> c#(a(a(a(x1))))
TRS:
a(x1) -> b(c(x1))
a(b(b(x1))) -> c(a(a(a(x1))))
c(c(x1)) -> x1
TDG Processor:
DPs:
a#(x1) -> c#(x1)
a#(b(b(x1))) -> a#(x1)
a#(b(b(x1))) -> a#(a(x1))
a#(b(b(x1))) -> a#(a(a(x1)))
a#(b(b(x1))) -> c#(a(a(a(x1))))
TRS:
a(x1) -> b(c(x1))
a(b(b(x1))) -> c(a(a(a(x1))))
c(c(x1)) -> x1
graph:
a#(b(b(x1))) -> a#(a(a(x1))) -> a#(b(b(x1))) -> c#(a(a(a(x1))))
a#(b(b(x1))) -> a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1)))
a#(b(b(x1))) -> a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(x1))
a#(b(b(x1))) -> a#(a(a(x1))) -> a#(b(b(x1))) -> a#(x1)
a#(b(b(x1))) -> a#(a(a(x1))) -> a#(x1) -> c#(x1)
a#(b(b(x1))) -> a#(a(x1)) -> a#(b(b(x1))) -> c#(a(a(a(x1))))
a#(b(b(x1))) -> a#(a(x1)) -> a#(b(b(x1))) -> a#(a(a(x1)))
a#(b(b(x1))) -> a#(a(x1)) -> a#(b(b(x1))) -> a#(a(x1))
a#(b(b(x1))) -> a#(a(x1)) -> a#(b(b(x1))) -> a#(x1)
a#(b(b(x1))) -> a#(a(x1)) -> a#(x1) -> c#(x1)
a#(b(b(x1))) -> a#(x1) -> a#(b(b(x1))) -> c#(a(a(a(x1))))
a#(b(b(x1))) -> a#(x1) -> a#(b(b(x1))) -> a#(a(a(x1)))
a#(b(b(x1))) -> a#(x1) -> a#(b(b(x1))) -> a#(a(x1))
a#(b(b(x1))) -> a#(x1) -> a#(b(b(x1))) -> a#(x1)
a#(b(b(x1))) -> a#(x1) -> a#(x1) -> c#(x1)
SCC Processor:
#sccs: 1
#rules: 3
#arcs: 15/25
DPs:
a#(b(b(x1))) -> a#(a(a(x1)))
a#(b(b(x1))) -> a#(x1)
a#(b(b(x1))) -> a#(a(x1))
TRS:
a(x1) -> b(c(x1))
a(b(b(x1))) -> c(a(a(a(x1))))
c(c(x1)) -> x1
Arctic Interpretation Processor:
dimension: 2
usable rules:
a(x1) -> b(c(x1))
a(b(b(x1))) -> c(a(a(a(x1))))
c(c(x1)) -> x1
interpretation:
[a#](x0) = [-& 0 ]x0 + [0],
[-& 2 ] [2]
[b](x0) = [0 0 ]x0 + [0],
[0 0 ] [0]
[c](x0) = [-& 0 ]x0 + [0],
[0 2] [2]
[a](x0) = [0 0]x0 + [0]
orientation:
a#(b(b(x1))) = [0 2]x1 + [2] >= [0 2]x1 + [2] = a#(a(a(x1)))
a#(b(b(x1))) = [0 2]x1 + [2] >= [-& 0 ]x1 + [0] = a#(x1)
a#(b(b(x1))) = [0 2]x1 + [2] >= [0 0]x1 + [0] = a#(a(x1))
[0 2] [2] [-& 2 ] [2]
a(x1) = [0 0]x1 + [0] >= [0 0 ]x1 + [0] = b(c(x1))
[2 4] [4] [2 4] [4]
a(b(b(x1))) = [2 2]x1 + [2] >= [2 2]x1 + [2] = c(a(a(a(x1))))
[0 0 ] [0]
c(c(x1)) = [-& 0 ]x1 + [0] >= x1 = x1
problem:
DPs:
a#(b(b(x1))) -> a#(a(a(x1)))
a#(b(b(x1))) -> a#(a(x1))
TRS:
a(x1) -> b(c(x1))
a(b(b(x1))) -> c(a(a(a(x1))))
c(c(x1)) -> x1
Restore Modifier:
DPs:
a#(b(b(x1))) -> a#(a(a(x1)))
a#(b(b(x1))) -> a#(a(x1))
TRS:
a(x1) -> b(c(x1))
a(b(b(x1))) -> c(a(a(a(x1))))
c(c(x1)) -> x1
EDG Processor:
DPs:
a#(b(b(x1))) -> a#(a(a(x1)))
a#(b(b(x1))) -> a#(a(x1))
TRS:
a(x1) -> b(c(x1))
a(b(b(x1))) -> c(a(a(a(x1))))
c(c(x1)) -> x1
graph:
a#(b(b(x1))) -> a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(x1))
a#(b(b(x1))) -> a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1)))
a#(b(b(x1))) -> a#(a(x1)) -> a#(b(b(x1))) -> a#(a(x1))
a#(b(b(x1))) -> a#(a(x1)) -> a#(b(b(x1))) -> a#(a(a(x1)))
Arctic Interpretation Processor:
dimension: 2
usable rules:
a(x1) -> b(c(x1))
a(b(b(x1))) -> c(a(a(a(x1))))
c(c(x1)) -> x1
interpretation:
[a#](x0) = [0 -&]x0 + [0],
[0 2] [0]
[b](x0) = [0 2]x0 + [2],
[0 0 ] [0 ]
[c](x0) = [-& 0 ]x0 + [-&],
[0 2] [0]
[a](x0) = [0 2]x0 + [2]
orientation:
a#(b(b(x1))) = [2 4]x1 + [4] >= [2 4]x1 + [4] = a#(a(a(x1)))
a#(b(b(x1))) = [2 4]x1 + [4] >= [0 2]x1 + [0] = a#(a(x1))
[0 2] [0] [0 2] [0]
a(x1) = [0 2]x1 + [2] >= [0 2]x1 + [2] = b(c(x1))
[4 6] [6] [4 6] [6]
a(b(b(x1))) = [4 6]x1 + [6] >= [4 6]x1 + [6] = c(a(a(a(x1))))
[0 0 ] [0 ]
c(c(x1)) = [-& 0 ]x1 + [-&] >= x1 = x1
problem:
DPs:
a#(b(b(x1))) -> a#(a(a(x1)))
TRS:
a(x1) -> b(c(x1))
a(b(b(x1))) -> c(a(a(a(x1))))
c(c(x1)) -> x1
Restore Modifier:
DPs:
a#(b(b(x1))) -> a#(a(a(x1)))
TRS:
a(x1) -> b(c(x1))
a(b(b(x1))) -> c(a(a(a(x1))))
c(c(x1)) -> x1
EDG Processor:
DPs:
a#(b(b(x1))) -> a#(a(a(x1)))
TRS:
a(x1) -> b(c(x1))
a(b(b(x1))) -> c(a(a(a(x1))))
c(c(x1)) -> x1
graph:
a#(b(b(x1))) -> a#(a(a(x1))) -> a#(b(b(x1))) -> a#(a(a(x1)))
Loop Processor:
loop length: 17
terms:
a(a(a(b(b(c(a(a(c(c(c(c(c(c(c(x1)))))))))))))))
a(a(c(a(a(a(c(a(a(c(c(c(c(c(c(c(x1))))))))))))))))
a(a(c(b(c(a(a(c(a(a(c(c(c(c(c(c(c(x1)))))))))))))))))
a(a(c(b(c(a(b(c(c(a(a(c(c(c(c(c(c(c(x1))))))))))))))))))
a(a(c(b(c(a(b(c(c(b(c(a(c(c(c(c(c(c(c(x1)))))))))))))))))))
a(a(c(b(c(a(b(b(c(a(c(c(c(c(c(c(c(x1)))))))))))))))))
a(b(c(c(b(c(a(b(b(c(a(c(c(c(c(c(c(c(x1))))))))))))))))))
a(b(c(c(b(c(c(a(a(a(c(a(c(c(c(c(c(c(c(x1)))))))))))))))))))
a(b(c(c(b(c(c(b(c(a(a(c(a(c(c(c(c(c(c(c(x1))))))))))))))))))))
a(b(b(c(c(b(c(a(a(c(a(c(c(c(c(c(c(c(x1))))))))))))))))))
a(b(b(c(c(b(c(a(b(c(c(a(c(c(c(c(c(c(c(x1)))))))))))))))))))
a(b(b(c(c(b(c(a(b(c(c(b(c(c(c(c(c(c(c(c(x1))))))))))))))))))))
a(b(b(b(c(a(b(c(c(b(c(c(c(c(c(c(c(c(x1))))))))))))))))))
c(a(a(a(b(c(a(b(c(c(b(c(c(c(c(c(c(c(c(x1)))))))))))))))))))
c(a(a(a(b(c(a(b(b(c(c(c(c(c(c(c(c(x1)))))))))))))))))
c(a(a(a(b(c(c(a(a(a(c(c(c(c(c(c(c(c(x1))))))))))))))))))
c(a(a(a(b(c(c(b(c(a(a(c(c(c(c(c(c(c(c(x1)))))))))))))))))))
context: c([])
substitution:
x1 -> c(x1)
Qed