Certification Problem

Input (TPDB SRS_Standard/Waldmann_07_size12/size-12-alpha-2-num-16)

The rewrite relation of the following TRS is considered.

a(x1) → x1 (1)
a(a(x1)) → b(x1) (2)
b(a(b(x1))) → a(a(b(b(b(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#(a(x1)) → b#(x1) (4)
b#(a(b(x1))) → a#(a(b(b(b(x1))))) (5)
b#(a(b(x1))) → a#(b(b(b(x1)))) (6)
b#(a(b(x1))) → b#(b(b(x1))) (7)
b#(a(b(x1))) → b#(b(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 0
-∞ -∞ -∞
-∞ -∞ -∞
· x1
[a(x1)] =
0
0
-∞
+
0 0 1
-∞ 0 0
0 -∞ 0
· x1
[b#(x1)] =
-∞
-∞
-∞
+
0 -∞ 0
-∞ -∞ -∞
-∞ -∞ -∞
· x1
[b(x1)] =
-∞
-∞
-∞
+
0 -∞ -∞
0 -∞ 0
0 0 -∞
· x1
the pair
b#(a(b(x1))) → b#(b(x1)) (8)
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
-∞ -∞ -∞
-∞ -∞ -∞
· x1
[a(x1)] =
-∞
-∞
0
+
0 1 -∞
0 0 0
-∞ 0 0
· x1
[b#(x1)] =
0
-∞
-∞
+
0 0 -∞
-∞ -∞ -∞
-∞ -∞ -∞
· x1
[b(x1)] =
-∞
-∞
-∞
+
0 0 -∞
0 0 0
-∞ 0 0
· x1
the pair
b#(a(b(x1))) → a#(b(b(b(x1)))) (6)
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
-∞
-∞
+
-∞ -∞ 1
-∞ -∞ -∞
-∞ -∞ -∞
· x1
[a(x1)] =
0
0
-∞
+
0 -∞ 0
0 0 -∞
1 -∞ 0
· x1
[b#(x1)] =
0
-∞
-∞
+
1 -∞ -∞
-∞ -∞ -∞
-∞ -∞ -∞
· x1
[b(x1)] =
-∞
-∞
0
+
0 -∞ -∞
0 0 0
1 -∞ -∞
· x1
the pair
b#(a(b(x1))) → b#(b(b(x1))) (7)
could be deleted.

1.1.1.1.1 Reduction Pair Processor

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

1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals
[b#(x1)] = 1 + 1 · x1
[a(x1)] = 1
[b(x1)] = 1 · x1
[a#(x1)] = 0
having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the pair
b#(a(b(x1))) → a#(a(b(b(b(x1))))) (5)
could be deleted.

1.1.1.1.1.1.1 P is empty

There are no pairs anymore.