YES Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

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

Proof

1 Rule Removal

Using the linear polynomial interpretation over the arctic semiring over the integers
[c(x1)] = 0 · x1 + -∞
[d(x1)] = 12 · x1 + -∞
[a(x1)] = 0 · x1 + -∞
[b(x1)] = 0 · x1 + -∞
the rules
b(c(a(x0))) → a(b(a(b(c(x0)))))
b(x0) → c(c(x0))
a(a(x0)) → a(c(b(a(x0))))
remain.

1.1 String Reversal

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

1.1.1 Bounds

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