YES Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

a(p(x0)) → p(a(A(x0)))
a(A(x0)) → A(a(x0))
p(A(A(x0))) → a(p(x0))

Proof

1 String Reversal

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

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 1
0 1 0
· x1 +
0 0 0
0 0 0
1 0 0
[p(x1)] =
1 1 1
0 1 1
0 1 1
· x1 +
1 0 0
1 0 0
0 0 0
[A(x1)] =
1 0 0
0 0 1
0 1 0
· x1 +
1 0 0
0 0 0
1 0 0
the rules
p(a(x0)) → A(a(p(x0)))
A(a(x0)) → a(A(x0))
remain.

1.1.1 String Reversal

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

1.1.1.1 Bounds

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