YES Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

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

Proof

1 Rule Removal

Using the linear polynomial interpretation over the naturals
[a(x1)] = 1 · x1 + 4
[c(x1)] = 2 · x1 + 0
[d(x1)] = 2 · x1 + 0
[b(x1)] = 1 · x1 + 1
the rules
a(a(d(d(x0)))) d(d(b(b(x0))))
c(c(x0)) d(d(x0))
remain.

1.1 Rule Removal

Using the linear polynomial interpretation over the arctic semiring over the integers
[a(x1)] = 12 · x1 + -∞
[c(x1)] = 0 · x1 + -∞
[d(x1)] = 0 · x1 + -∞
[b(x1)] = 7 · x1 + -∞
the rule
c(c(x0)) d(d(x0))
remains.

1.1.1 Rule Removal

Using the linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1 over the arctic semiring over the integers
[c(x1)] =
0 2
3 0
· x1 +
-∞ -∞
-∞ -∞
[d(x1)] =
0 0
0 0
· x1 +
-∞ -∞
-∞ -∞
all rules could be removed.

1.1.1.1 R is empty

There are no rules in the TRS. Hence, it is terminating.