YES Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

Begin(c(a(b(b(c(x0)))))) → Wait(Right1(x0))
Begin(a(b(b(c(x0))))) → Wait(Right2(x0))
Begin(b(b(c(x0)))) → Wait(Right3(x0))
Begin(b(c(x0))) → Wait(Right4(x0))
Begin(c(x0)) → Wait(Right5(x0))
Right1(b(End(x0))) → Left(a(b(c(b(b(c(a(End(x0)))))))))
Right2(b(c(End(x0)))) → Left(a(b(c(b(b(c(a(End(x0)))))))))
Right3(b(c(a(End(x0))))) → Left(a(b(c(b(b(c(a(End(x0)))))))))
Right4(b(c(a(b(End(x0)))))) → Left(a(b(c(b(b(c(a(End(x0)))))))))
Right5(b(c(a(b(b(End(x0))))))) → Left(a(b(c(b(b(c(a(End(x0)))))))))
Right1(b(x0)) → Ab(Right1(x0))
Right2(b(x0)) → Ab(Right2(x0))
Right3(b(x0)) → Ab(Right3(x0))
Right4(b(x0)) → Ab(Right4(x0))
Right5(b(x0)) → Ab(Right5(x0))
Right1(c(x0)) → Ac(Right1(x0))
Right2(c(x0)) → Ac(Right2(x0))
Right3(c(x0)) → Ac(Right3(x0))
Right4(c(x0)) → Ac(Right4(x0))
Right5(c(x0)) → Ac(Right5(x0))
Right1(a(x0)) → Aa(Right1(x0))
Right2(a(x0)) → Aa(Right2(x0))
Right3(a(x0)) → Aa(Right3(x0))
Right4(a(x0)) → Aa(Right4(x0))
Right5(a(x0)) → Aa(Right5(x0))
Ab(Left(x0)) → Left(b(x0))
Ac(Left(x0)) → Left(c(x0))
Aa(Left(x0)) → Left(a(x0))
Wait(Left(x0)) → Begin(x0)
b(c(a(b(b(c(x0)))))) → a(b(c(b(b(c(a(x0)))))))

Proof

1 Rule Removal

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals
[Right1(x1)] =
1 0 0
0 1 0
0 0 1
· x1 +
1 0 0
0 0 0
0 0 0
[Right4(x1)] =
1 0 0
0 1 0
0 0 1
· x1 +
0 0 0
0 0 0
0 0 0
[b(x1)] =
1 0 0
0 0 0
0 0 1
· x1 +
0 0 0
0 0 0
0 0 0
[Begin(x1)] =
1 1 1
1 1 1
1 1 1
· x1 +
0 0 0
0 0 0
0 0 0
[Wait(x1)] =
1 1 1
1 1 1
1 1 1
· x1 +
0 0 0
0 0 0
0 0 0
[End(x1)] =
1 0 0
0 0 0
0 0 0
· x1 +
0 0 0
0 0 0
0 0 0
[c(x1)] =
1 0 0
0 0 0
0 1 1
· x1 +
0 0 0
1 0 0
0 0 0
[Right3(x1)] =
1 0 0
0 1 0
0 0 1
· x1 +
0 0 0
0 0 0
0 0 0
[Aa(x1)] =
1 0 1
0 0 0
0 0 0
· x1 +
0 0 0
0 0 0
0 0 0
[Ab(x1)] =
1 0 0
0 0 0
0 0 1
· x1 +
0 0 0
0 0 0
0 0 0
[Left(x1)] =
1 0 0
0 1 0
0 0 1
· x1 +
0 0 0
0 0 0
0 0 0
[Right5(x1)] =
1 0 0
0 1 0
0 0 1
· x1 +
1 0 0
0 0 0
0 0 0
[Ac(x1)] =
1 0 0
0 0 0
0 1 1
· x1 +
0 0 0
1 0 0
0 0 0
[Right2(x1)] =
1 0 0
0 1 0
0 0 1
· x1 +
0 0 0
0 0 0
0 0 0
[a(x1)] =
1 0 1
0 0 0
0 0 0
· x1 +
0 0 0
0 0 0
0 0 0
the rules
Begin(c(a(b(b(c(x0)))))) → Wait(Right1(x0))
Begin(a(b(b(c(x0))))) → Wait(Right2(x0))
Begin(b(b(c(x0)))) → Wait(Right3(x0))
Begin(b(c(x0))) → Wait(Right4(x0))
Begin(c(x0)) → Wait(Right5(x0))
Right2(b(c(End(x0)))) → Left(a(b(c(b(b(c(a(End(x0)))))))))
Right3(b(c(a(End(x0))))) → Left(a(b(c(b(b(c(a(End(x0)))))))))
Right4(b(c(a(b(End(x0)))))) → Left(a(b(c(b(b(c(a(End(x0)))))))))
Right1(b(x0)) → Ab(Right1(x0))
Right2(b(x0)) → Ab(Right2(x0))
Right3(b(x0)) → Ab(Right3(x0))
Right4(b(x0)) → Ab(Right4(x0))
Right5(b(x0)) → Ab(Right5(x0))
Right1(c(x0)) → Ac(Right1(x0))
Right2(c(x0)) → Ac(Right2(x0))
Right3(c(x0)) → Ac(Right3(x0))
Right4(c(x0)) → Ac(Right4(x0))
Right5(c(x0)) → Ac(Right5(x0))
Right1(a(x0)) → Aa(Right1(x0))
Right2(a(x0)) → Aa(Right2(x0))
Right3(a(x0)) → Aa(Right3(x0))
Right4(a(x0)) → Aa(Right4(x0))
Right5(a(x0)) → Aa(Right5(x0))
Ab(Left(x0)) → Left(b(x0))
Ac(Left(x0)) → Left(c(x0))
Aa(Left(x0)) → Left(a(x0))
Wait(Left(x0)) → Begin(x0)
b(c(a(b(b(c(x0)))))) → a(b(c(b(b(c(a(x0)))))))
remain.

1.1 Rule Removal

Using the linear polynomial interpretation over the arctic semiring over the integers
[Right1(x1)] = 0 · x1 + -∞
[Right4(x1)] = 3 · x1 + -∞
[b(x1)] = 0 · x1 + -∞
[Begin(x1)] = 3 · x1 + -∞
[Wait(x1)] = 0 · x1 + -∞
[End(x1)] = 0 · x1 + -∞
[c(x1)] = 0 · x1 + -∞
[Right3(x1)] = 3 · x1 + -∞
[Aa(x1)] = 0 · x1 + -∞
[Ab(x1)] = 0 · x1 + -∞
[Left(x1)] = 3 · x1 + -∞
[Right5(x1)] = 2 · x1 + -∞
[Ac(x1)] = 0 · x1 + -∞
[Right2(x1)] = 3 · x1 + -∞
[a(x1)] = 0 · x1 + -∞
the rules
Begin(a(b(b(c(x0))))) → Wait(Right2(x0))
Begin(b(b(c(x0)))) → Wait(Right3(x0))
Begin(b(c(x0))) → Wait(Right4(x0))
Right2(b(c(End(x0)))) → Left(a(b(c(b(b(c(a(End(x0)))))))))
Right3(b(c(a(End(x0))))) → Left(a(b(c(b(b(c(a(End(x0)))))))))
Right4(b(c(a(b(End(x0)))))) → Left(a(b(c(b(b(c(a(End(x0)))))))))
Right1(b(x0)) → Ab(Right1(x0))
Right2(b(x0)) → Ab(Right2(x0))
Right3(b(x0)) → Ab(Right3(x0))
Right4(b(x0)) → Ab(Right4(x0))
Right5(b(x0)) → Ab(Right5(x0))
Right1(c(x0)) → Ac(Right1(x0))
Right2(c(x0)) → Ac(Right2(x0))
Right3(c(x0)) → Ac(Right3(x0))
Right4(c(x0)) → Ac(Right4(x0))
Right5(c(x0)) → Ac(Right5(x0))
Right1(a(x0)) → Aa(Right1(x0))
Right2(a(x0)) → Aa(Right2(x0))
Right3(a(x0)) → Aa(Right3(x0))
Right4(a(x0)) → Aa(Right4(x0))
Right5(a(x0)) → Aa(Right5(x0))
Ab(Left(x0)) → Left(b(x0))
Ac(Left(x0)) → Left(c(x0))
Aa(Left(x0)) → Left(a(x0))
Wait(Left(x0)) → Begin(x0)
b(c(a(b(b(c(x0)))))) → a(b(c(b(b(c(a(x0)))))))
remain.

1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[Right1(x1)] = 4 · x1 + 0
[Right4(x1)] = 4 · x1 + 1
[b(x1)] = 1 · x1 + 0
[Begin(x1)] = 1 · x1 + 7
[Wait(x1)] = 1 · x1 + 7
[End(x1)] = 1 · x1 + 0
[c(x1)] = 4 · x1 + 1
[Right3(x1)] = 4 · x1 + 1
[Aa(x1)] = 1 · x1 + 0
[Ab(x1)] = 1 · x1 + 0
[Left(x1)] = 1 · x1 + 0
[Right5(x1)] = 4 · x1 + 1
[Ac(x1)] = 4 · x1 + 1
[Right2(x1)] = 4 · x1 + 1
[a(x1)] = 1 · x1 + 0
the rules
Begin(a(b(b(c(x0))))) → Wait(Right2(x0))
Begin(b(b(c(x0)))) → Wait(Right3(x0))
Begin(b(c(x0))) → Wait(Right4(x0))
Right2(b(c(End(x0)))) → Left(a(b(c(b(b(c(a(End(x0)))))))))
Right3(b(c(a(End(x0))))) → Left(a(b(c(b(b(c(a(End(x0)))))))))
Right4(b(c(a(b(End(x0)))))) → Left(a(b(c(b(b(c(a(End(x0)))))))))
Right1(b(x0)) → Ab(Right1(x0))
Right2(b(x0)) → Ab(Right2(x0))
Right3(b(x0)) → Ab(Right3(x0))
Right4(b(x0)) → Ab(Right4(x0))
Right5(b(x0)) → Ab(Right5(x0))
Right2(c(x0)) → Ac(Right2(x0))
Right3(c(x0)) → Ac(Right3(x0))
Right4(c(x0)) → Ac(Right4(x0))
Right5(c(x0)) → Ac(Right5(x0))
Right1(a(x0)) → Aa(Right1(x0))
Right2(a(x0)) → Aa(Right2(x0))
Right3(a(x0)) → Aa(Right3(x0))
Right4(a(x0)) → Aa(Right4(x0))
Right5(a(x0)) → Aa(Right5(x0))
Ab(Left(x0)) → Left(b(x0))
Ac(Left(x0)) → Left(c(x0))
Aa(Left(x0)) → Left(a(x0))
Wait(Left(x0)) → Begin(x0)
b(c(a(b(b(c(x0)))))) → a(b(c(b(b(c(a(x0)))))))
remain.

1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[Right1(x1)] = 5 · x1 + 3
[Right4(x1)] = 2 · x1 + 1
[b(x1)] = 2 · x1 + 1
[Begin(x1)] = 1 · x1 + 0
[Wait(x1)] = 1 · x1 + 0
[End(x1)] = 2 · x1 + 0
[c(x1)] = 1 · x1 + 0
[Right3(x1)] = 4 · x1 + 3
[Aa(x1)] = 1 · x1 + 0
[Ab(x1)] = 2 · x1 + 1
[Left(x1)] = 1 · x1 + 0
[Right5(x1)] = 1 · x1 + 0
[Ac(x1)] = 1 · x1 + 0
[Right2(x1)] = 4 · x1 + 3
[a(x1)] = 1 · x1 + 0
the rules
Begin(a(b(b(c(x0))))) → Wait(Right2(x0))
Begin(b(b(c(x0)))) → Wait(Right3(x0))
Begin(b(c(x0))) → Wait(Right4(x0))
Right2(b(c(End(x0)))) → Left(a(b(c(b(b(c(a(End(x0)))))))))
Right3(b(c(a(End(x0))))) → Left(a(b(c(b(b(c(a(End(x0)))))))))
Right4(b(c(a(b(End(x0)))))) → Left(a(b(c(b(b(c(a(End(x0)))))))))
Right2(b(x0)) → Ab(Right2(x0))
Right3(b(x0)) → Ab(Right3(x0))
Right4(b(x0)) → Ab(Right4(x0))
Right5(b(x0)) → Ab(Right5(x0))
Right2(c(x0)) → Ac(Right2(x0))
Right3(c(x0)) → Ac(Right3(x0))
Right4(c(x0)) → Ac(Right4(x0))
Right5(c(x0)) → Ac(Right5(x0))
Right1(a(x0)) → Aa(Right1(x0))
Right2(a(x0)) → Aa(Right2(x0))
Right3(a(x0)) → Aa(Right3(x0))
Right4(a(x0)) → Aa(Right4(x0))
Right5(a(x0)) → Aa(Right5(x0))
Ab(Left(x0)) → Left(b(x0))
Ac(Left(x0)) → Left(c(x0))
Aa(Left(x0)) → Left(a(x0))
Wait(Left(x0)) → Begin(x0)
b(c(a(b(b(c(x0)))))) → a(b(c(b(b(c(a(x0)))))))
remain.

1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[Right1(x1)] = 8 · x1 + 0
[Right4(x1)] = 4 · x1 + 2
[b(x1)] = 2 · x1 + 1
[Begin(x1)] = 2 · x1 + 0
[Wait(x1)] = 1 · x1 + 0
[End(x1)] = 1 · x1 + 0
[c(x1)] = 1 · x1 + 0
[Right3(x1)] = 8 · x1 + 6
[Aa(x1)] = 1 · x1 + 0
[Ab(x1)] = 2 · x1 + 2
[Left(x1)] = 2 · x1 + 0
[Right5(x1)] = 14 · x1 + 0
[Ac(x1)] = 1 · x1 + 0
[Right2(x1)] = 8 · x1 + 6
[a(x1)] = 1 · x1 + 0
the rules
Begin(a(b(b(c(x0))))) → Wait(Right2(x0))
Begin(b(b(c(x0)))) → Wait(Right3(x0))
Begin(b(c(x0))) → Wait(Right4(x0))
Right2(b(c(End(x0)))) → Left(a(b(c(b(b(c(a(End(x0)))))))))
Right3(b(c(a(End(x0))))) → Left(a(b(c(b(b(c(a(End(x0)))))))))
Right4(b(c(a(b(End(x0)))))) → Left(a(b(c(b(b(c(a(End(x0)))))))))
Right2(b(x0)) → Ab(Right2(x0))
Right3(b(x0)) → Ab(Right3(x0))
Right4(b(x0)) → Ab(Right4(x0))
Right2(c(x0)) → Ac(Right2(x0))
Right3(c(x0)) → Ac(Right3(x0))
Right4(c(x0)) → Ac(Right4(x0))
Right5(c(x0)) → Ac(Right5(x0))
Right1(a(x0)) → Aa(Right1(x0))
Right2(a(x0)) → Aa(Right2(x0))
Right3(a(x0)) → Aa(Right3(x0))
Right4(a(x0)) → Aa(Right4(x0))
Right5(a(x0)) → Aa(Right5(x0))
Ab(Left(x0)) → Left(b(x0))
Ac(Left(x0)) → Left(c(x0))
Aa(Left(x0)) → Left(a(x0))
Wait(Left(x0)) → Begin(x0)
b(c(a(b(b(c(x0)))))) → a(b(c(b(b(c(a(x0)))))))
remain.

1.1.1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the naturals
[Right1(x1)] = 1 · x1 + 3
[Right4(x1)] = 4 · x1 + 4
[b(x1)] = 1 · x1 + 0
[Begin(x1)] = 2 · x1 + 0
[Wait(x1)] = 2 · x1 + 0
[End(x1)] = 1 · x1 + 0
[c(x1)] = 4 · x1 + 4
[Right3(x1)] = 4 · x1 + 4
[Aa(x1)] = 1 · x1 + 0
[Ab(x1)] = 1 · x1 + 0
[Left(x1)] = 1 · x1 + 0
[Right5(x1)] = 2 · x1 + 0
[Ac(x1)] = 4 · x1 + 4
[Right2(x1)] = 4 · x1 + 4
[a(x1)] = 1 · x1 + 0
the rules
Begin(a(b(b(c(x0))))) → Wait(Right2(x0))
Begin(b(b(c(x0)))) → Wait(Right3(x0))
Begin(b(c(x0))) → Wait(Right4(x0))
Right2(b(c(End(x0)))) → Left(a(b(c(b(b(c(a(End(x0)))))))))
Right3(b(c(a(End(x0))))) → Left(a(b(c(b(b(c(a(End(x0)))))))))
Right4(b(c(a(b(End(x0)))))) → Left(a(b(c(b(b(c(a(End(x0)))))))))
Right2(b(x0)) → Ab(Right2(x0))
Right3(b(x0)) → Ab(Right3(x0))
Right4(b(x0)) → Ab(Right4(x0))
Right2(c(x0)) → Ac(Right2(x0))
Right3(c(x0)) → Ac(Right3(x0))
Right4(c(x0)) → Ac(Right4(x0))
Right1(a(x0)) → Aa(Right1(x0))
Right2(a(x0)) → Aa(Right2(x0))
Right3(a(x0)) → Aa(Right3(x0))
Right4(a(x0)) → Aa(Right4(x0))
Right5(a(x0)) → Aa(Right5(x0))
Ab(Left(x0)) → Left(b(x0))
Ac(Left(x0)) → Left(c(x0))
Aa(Left(x0)) → Left(a(x0))
Wait(Left(x0)) → Begin(x0)
b(c(a(b(b(c(x0)))))) → a(b(c(b(b(c(a(x0)))))))
remain.

1.1.1.1.1.1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.
Begin#(a(b(b(c(x0))))) → Right2#(x0)
Begin#(a(b(b(c(x0))))) → Wait#(Right2(x0))
Begin#(b(b(c(x0)))) → Right3#(x0)
Begin#(b(b(c(x0)))) → Wait#(Right3(x0))
Begin#(b(c(x0))) → Right4#(x0)
Begin#(b(c(x0))) → Wait#(Right4(x0))
Right2#(b(c(End(x0)))) → b#(c(a(End(x0))))
Right2#(b(c(End(x0)))) → b#(b(c(a(End(x0)))))
Right2#(b(c(End(x0)))) → b#(c(b(b(c(a(End(x0)))))))
Right3#(b(c(a(End(x0))))) → b#(b(c(a(End(x0)))))
Right3#(b(c(a(End(x0))))) → b#(c(b(b(c(a(End(x0)))))))
Right4#(b(c(a(b(End(x0)))))) → b#(c(a(End(x0))))
Right4#(b(c(a(b(End(x0)))))) → b#(b(c(a(End(x0)))))
Right4#(b(c(a(b(End(x0)))))) → b#(c(b(b(c(a(End(x0)))))))
Right2#(b(x0)) → Right2#(x0)
Right2#(b(x0)) → Ab#(Right2(x0))
Right3#(b(x0)) → Right3#(x0)
Right3#(b(x0)) → Ab#(Right3(x0))
Right4#(b(x0)) → Right4#(x0)
Right4#(b(x0)) → Ab#(Right4(x0))
Right2#(c(x0)) → Right2#(x0)
Right2#(c(x0)) → Ac#(Right2(x0))
Right3#(c(x0)) → Right3#(x0)
Right3#(c(x0)) → Ac#(Right3(x0))
Right4#(c(x0)) → Right4#(x0)
Right4#(c(x0)) → Ac#(Right4(x0))
Right1#(a(x0)) → Right1#(x0)
Right1#(a(x0)) → Aa#(Right1(x0))
Right2#(a(x0)) → Right2#(x0)
Right2#(a(x0)) → Aa#(Right2(x0))
Right3#(a(x0)) → Right3#(x0)
Right3#(a(x0)) → Aa#(Right3(x0))
Right4#(a(x0)) → Right4#(x0)
Right4#(a(x0)) → Aa#(Right4(x0))
Right5#(a(x0)) → Right5#(x0)
Right5#(a(x0)) → Aa#(Right5(x0))
Ab#(Left(x0)) → b#(x0)
Wait#(Left(x0)) → Begin#(x0)
b#(c(a(b(b(c(x0)))))) → b#(c(a(x0)))
b#(c(a(b(b(c(x0)))))) → b#(b(c(a(x0))))
b#(c(a(b(b(c(x0)))))) → b#(c(b(b(c(a(x0))))))

1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.