YES Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

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

Proof

1 Rule Removal

Using the linear polynomial interpretation over the arctic semiring over the integers
[d(x1)] = 0 · x1 + -∞
[a(x1)] = 1 · x1 + -∞
[b(x1)] = 1 · x1 + -∞
[c(x1)] = 1 · x1 + -∞
the rules
b(b(d(d(b(b(x0)))))) → c(c(d(d(b(b(x0))))))
a(a(d(d(x0)))) → d(d(c(c(x0))))
b(b(b(b(b(b(x0)))))) → a(a(b(b(c(c(x0))))))
d(d(c(c(x0)))) → b(b(d(d(x0))))
d(d(c(c(x0)))) → d(d(b(b(d(d(x0))))))
d(d(a(a(c(c(x0)))))) → b(b(b(b(x0))))
remain.

1.1 Rule Removal

Using the linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1 over the arctic semiring over the integers
[d(x1)] =
0 0
0 0
· x1 +
-∞ -∞
-∞ -∞
[a(x1)] =
0 0
-∞ 1
· x1 +
-∞ -∞
-∞ -∞
[b(x1)] =
0 0
-∞ 0
· x1 +
-∞ -∞
-∞ -∞
[c(x1)] =
0 0
-∞ -∞
· x1 +
-∞ -∞
-∞ -∞
the rules
b(b(d(d(b(b(x0)))))) → c(c(d(d(b(b(x0))))))
b(b(b(b(b(b(x0)))))) → a(a(b(b(c(c(x0))))))
d(d(c(c(x0)))) → b(b(d(d(x0))))
d(d(c(c(x0)))) → d(d(b(b(d(d(x0))))))
d(d(a(a(c(c(x0)))))) → b(b(b(b(x0))))
remain.

1.1.1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS
b(b(d(d(b(b(x0)))))) → b(b(d(d(c(c(x0))))))
b(b(b(b(b(b(x0)))))) → c(c(b(b(a(a(x0))))))
c(c(d(d(x0)))) → d(d(b(b(x0))))
c(c(d(d(x0)))) → d(d(b(b(d(d(x0))))))
c(c(a(a(d(d(x0)))))) → b(b(b(b(x0))))

1.1.1.1 Bounds

The given TRS is match-bounded by 1. This is shown by the following automaton.