YES Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

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

Proof

1 Rule Removal

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

1.1 String Reversal

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

1.1.1 Dependency Pair Transformation

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

1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the arctic semiring over the integers
[a#(x1)] = 4 · x1 + -∞
[b(x1)] = 4 · x1 + -∞
[b#(x1)] = 0 · x1 + -∞
[a(x1)] = 4 · x1 + -∞
[c#(x1)] = 8 · x1 + -∞
[c(x1)] = 8 · x1 + -∞
together with the usable rules
a(b(a(x0))) → c(b(x0))
b(b(b(x0))) → b(c(x0))
c(x0) → b(a(x0))
(w.r.t. the implicit argument filter of the reduction pair), the pairs
a#(b(a(x0))) → c#(b(x0))
b#(b(b(x0))) → c#(x0)
b#(b(b(x0))) → b#(c(x0))
remain.

1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.