YES Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

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

Proof

1 Rule Removal

Using the linear polynomial interpretation over the arctic semiring over the integers
[c(x1)] = 12 · x1 + -∞
[d(x1)] = 8 · x1 + -∞
[a(x1)] = 12 · x1 + -∞
[b(x1)] = 12 · x1 + -∞
the rules
a(a(x0)) → b(c(x0))
c(c(x0)) → d(d(d(x0)))
d(d(d(x0))) → a(c(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
[c(x1)] =
0 0
2 2
· x1 +
-∞ -∞
-∞ -∞
[d(x1)] =
0 0
1 1
· x1 +
-∞ -∞
-∞ -∞
[a(x1)] =
0 0
1 1
· x1 +
-∞ -∞
-∞ -∞
[b(x1)] =
0 -∞
-∞ -∞
· x1 +
-∞ -∞
-∞ -∞
the rules
c(c(x0)) → d(d(d(x0)))
d(d(d(x0))) → a(c(x0))
remain.

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 0
1 2
· x1 +
-∞ -∞
-∞ -∞
[d(x1)] =
0 0
0 0
· x1 +
-∞ -∞
-∞ -∞
[a(x1)] =
0 -∞
-∞ -∞
· x1 +
-∞ -∞
-∞ -∞
the rule
d(d(d(x0))) → a(c(x0))
remains.

1.1.1.1 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 0
0 1
· x1 +
-∞ -∞
-∞ -∞
[d(x1)] =
0 0
1 1
· x1 +
-∞ -∞
-∞ -∞
[a(x1)] =
1 0
0 0
· x1 +
-∞ -∞
-∞ -∞
all rules could be removed.

1.1.1.1.1 R is empty

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