Certification Problem

Input (TPDB SRS_Standard/Waldmann_07_size12/size-12-alpha-3-num-293)

The rewrite relation of the following TRS is considered.

a(x1) → x1 (1)
a(b(b(x1))) → b(b(a(b(c(x1))))) (2)
b(c(x1)) → a(x1) (3)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.
a#(b(b(x1))) → b#(b(a(b(c(x1))))) (4)
a#(b(b(x1))) → b#(a(b(c(x1)))) (5)
a#(b(b(x1))) → a#(b(c(x1))) (6)
a#(b(b(x1))) → b#(c(x1)) (7)
b#(c(x1)) → a#(x1) (8)

1.1 Reduction Pair Processor

Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals
[a#(x1)] =
0
-∞
-∞
+
0 1 0
-∞ -∞ -∞
-∞ -∞ -∞
· x1
[b(x1)] =
-∞
-∞
-∞
+
0 -∞ -∞
0 -∞ 0
0 0 -∞
· x1
[b#(x1)] =
-∞
-∞
-∞
+
0 0 -∞
-∞ -∞ -∞
-∞ -∞ -∞
· x1
[a(x1)] =
-∞
-∞
0
+
0 0 -∞
-∞ 0 -∞
0 1 0
· x1
[c(x1)] =
-∞
0
-∞
+
0 0 -∞
0 1 0
-∞ 0 -∞
· x1
the pair
a#(b(b(x1))) → b#(a(b(c(x1)))) (5)
could be deleted.

1.1.1 Reduction Pair Processor

Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals
[a#(x1)] =
0
-∞
-∞
+
0 0 0
-∞ -∞ -∞
-∞ -∞ -∞
· x1
[b(x1)] =
1
0
0
+
0 0 0
0 0 0
1 0 0
· x1
[b#(x1)] =
-∞
-∞
-∞
+
0 0 0
-∞ -∞ -∞
-∞ -∞ -∞
· x1
[a(x1)] =
0
0
0
+
0 0 0
0 0 0
0 0 0
· x1
[c(x1)] =
0
0
0
+
-∞ -∞ -∞
0 -∞ 0
0 0 0
· x1
the pairs
a#(b(b(x1))) → a#(b(c(x1))) (6)
a#(b(b(x1))) → b#(c(x1)) (7)
could be deleted.

1.1.1.1 Reduction Pair Processor

Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals
[a#(x1)] =
0
-∞
-∞
+
0 0 0
-∞ -∞ -∞
-∞ -∞ -∞
· x1
[b(x1)] =
1
-∞
-∞
+
0 0 0
0 0 -∞
1 0 0
· x1
[b#(x1)] =
-∞
-∞
-∞
+
0 0 -∞
-∞ -∞ -∞
-∞ -∞ -∞
· x1
[a(x1)] =
0
0
0
+
0 0 0
0 0 -∞
-∞ -∞ 0
· x1
[c(x1)] =
0
0
-∞
+
-∞ -∞ -∞
0 0 0
-∞ 0 0
· x1
the pair
a#(b(b(x1))) → b#(b(a(b(c(x1))))) (4)
could be deleted.

1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals
[c(x1)] = 1 · x1
[a#(x1)] = 1 · x1
[b#(x1)] = 1 · x1
having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.

1.1.1.1.1.1 Size-Change Termination

Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

b#(c(x1)) → a#(x1) (8)
1 > 1

As there is no critical graph in the transitive closure, there are no infinite chains.