YES Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

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

Proof

1 Rule Removal

Using the linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1 over the naturals
[f(x1)] =
1 1
0 0
· x1 +
1 0
0 0
[d(x1)] =
1 1
0 0
· x1 +
0 0
2 0
[g(x1)] =
1 1
0 0
· x1 +
1 0
2 0
[a(x1)] =
1 0
0 0
· x1 +
0 0
0 0
[b(x1)] =
1 1
0 0
· x1 +
1 0
2 0
[c(x1)] =
1 1
0 0
· x1 +
1 0
2 0
the rules
c(c(x0)) → d(d(d(x0)))
c(x0) → g(x0)
d(d(x0)) → c(f(x0))
d(d(d(x0))) → g(c(x0))
f(x0) → a(g(x0))
g(x0) → d(a(b(x0)))
g(g(x0)) → b(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
[f(x1)] =
0 2
-∞ 0
· x1 +
-∞ -∞
-∞ -∞
[d(x1)] =
0 0
0 2
· x1 +
-∞ -∞
-∞ -∞
[g(x1)] =
0 0
1 3
· x1 +
-∞ -∞
-∞ -∞
[a(x1)] =
0 -∞
-∞ -∞
· x1 +
-∞ -∞
-∞ -∞
[b(x1)] =
0 -∞
-∞ -∞
· x1 +
-∞ -∞
-∞ -∞
[c(x1)] =
0 2
1 3
· x1 +
-∞ -∞
-∞ -∞
the rules
c(c(x0)) → d(d(d(x0)))
c(x0) → g(x0)
d(d(x0)) → c(f(x0))
d(d(d(x0))) → g(c(x0))
f(x0) → a(g(x0))
g(x0) → d(a(b(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
[f(x1)] =
0 -∞
-∞ -∞
· x1 +
-∞ -∞
-∞ -∞
[d(x1)] =
0 0
0 1
· x1 +
-∞ -∞
-∞ -∞
[g(x1)] =
0 -∞
0 -∞
· x1 +
-∞ -∞
-∞ -∞
[a(x1)] =
0 0
-∞ -∞
· x1 +
-∞ -∞
-∞ -∞
[b(x1)] =
0 -∞
0 -∞
· x1 +
-∞ -∞
-∞ -∞
[c(x1)] =
0 1
0 2
· x1 +
-∞ -∞
-∞ -∞
the rules
c(c(x0)) → d(d(d(x0)))
c(x0) → g(x0)
d(d(x0)) → c(f(x0))
f(x0) → a(g(x0))
g(x0) → d(a(b(x0)))
remain.

1.1.1.1 String Reversal

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

1.1.1.1.1 Bounds

The given TRS is match-bounded by 3. This is shown by the following automaton.