YES Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

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

Proof

1 Rule Removal

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

1.1 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 +
0 0
1 0
[a(x1)] =
1 0
0 2
· x1 +
0 0
0 0
[d(x1)] =
1 0
0 0
· x1 +
0 0
0 0
[e(x1)] =
2 1
0 0
· x1 +
0 0
1 0
[b(x1)] =
1 0
0 0
· x1 +
0 0
0 0
[c(x1)] =
1 0
0 0
· x1 +
0 0
0 0
the rules
a(b(x0)) → b(c(a(x0)))
b(c(x0)) → c(b(b(x0)))
a(c(x0)) → c(a(b(x0)))
a(a(x0)) → a(d(d(d(x0))))
d(a(x0)) → d(d(c(x0)))
a(d(d(c(x0)))) → a(a(a(d(x0))))
b(d(x0)) → d(d(x0))
remain.

1.1.1 String Reversal

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

1.1.1.1 Rule Removal

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

1.1.1.1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.
b#(a(x0)) → b#(x0)
b#(a(x0)) → c#(b(x0))
b#(a(x0)) → a#(c(b(x0)))
c#(b(x0)) → c#(x0)
c#(b(x0)) → b#(c(x0))
c#(b(x0)) → b#(b(c(x0)))
c#(a(x0)) → c#(x0)
c#(a(x0)) → a#(c(x0))
c#(a(x0)) → b#(a(c(x0)))
a#(d(x0)) → c#(d(d(x0)))
c#(d(d(a(x0)))) → a#(a(x0))
c#(d(d(a(x0)))) → a#(a(a(x0)))

1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.