YES Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

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

Proof

1 String Reversal

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

1.1 Rule Removal

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

1.1.1.1 Rule Removal

Using the linear polynomial interpretation over the arctic semiring over the integers
[e(x1)] = 2 · x1 + -∞
[a(x1)] = 5 · x1 + -∞
[c(x1)] = 14 · x1 + -∞
[b(x1)] = 2 · x1 + -∞
[d(x1)] = 4 · x1 + -∞
the rules
b(e(b(x0))) → d(e(x0))
e(e(e(x0))) → e(d(x0))
d(x0) → e(b(x0))
remain.

1.1.1.1.1 Dependency Pair Transformation

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

1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.