YES Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

a(a(b(b(x0)))) → b(b(c(c(a(a(x0))))))
b(b(c(c(x0)))) → c(c(b(b(b(b(x0))))))
b(b(a(a(x0)))) → a(a(c(c(b(b(x0))))))

Proof

1 Rule Removal

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals
[a(x1)] =
1 0 1
0 0 0
1 0 0
· x1 +
0 0 0
1 0 0
0 0 0
[b(x1)] =
1 0 0
0 0 0
0 1 1
· x1 +
0 0 0
1 0 0
0 0 0
[c(x1)] =
1 0 0
0 0 0
0 0 0
· x1 +
0 0 0
0 0 0
0 0 0
the rules
b(b(c(c(x0)))) → c(c(b(b(b(b(x0))))))
b(b(a(a(x0)))) → a(a(c(c(b(b(x0))))))
remain.

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.
b#(b(c(c(x0)))) → b#(x0)
b#(b(c(c(x0)))) → b#(b(x0))
b#(b(c(c(x0)))) → b#(b(b(x0)))
b#(b(c(c(x0)))) → b#(b(b(b(x0))))
b#(b(a(a(x0)))) → b#(x0)
b#(b(a(a(x0)))) → b#(b(x0))

1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the arctic semiring over the integers
[a(x1)] = 2 · x1 + -∞
[b#(x1)] = 0 · x1 + -∞
[b(x1)] = 0 · x1 + -∞
[c(x1)] = 0 · x1 + -∞
together with the usable rules
b(b(c(c(x0)))) → c(c(b(b(b(b(x0))))))
b(b(a(a(x0)))) → a(a(c(c(b(b(x0))))))
(w.r.t. the implicit argument filter of the reduction pair), the pairs
b#(b(c(c(x0)))) → b#(x0)
b#(b(c(c(x0)))) → b#(b(x0))
b#(b(c(c(x0)))) → b#(b(b(x0)))
b#(b(c(c(x0)))) → b#(b(b(b(x0))))
remain.

1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the arctic semiring over the integers
[a(x1)] = -∞ · x1 + 10
[b#(x1)] = 0 · x1 + 0
[b(x1)] = 0 · x1 + 4
[c(x1)] = 1 · x1 + 7
together with the usable rules
b(b(c(c(x0)))) → c(c(b(b(b(b(x0))))))
b(b(a(a(x0)))) → a(a(c(c(b(b(x0))))))
(w.r.t. the implicit argument filter of the reduction pair), all pairs could be removed.

1.1.1.1.1 P is empty

There are no pairs anymore.