Certification Problem

Input (TPDB SRS_Standard/Secret_07_SRS/x10)

The rewrite relation of the following TRS is considered.

a(a(b(c(x1)))) → b(b(a(a(x1)))) (1)
b(x1) → c(c(a(a(x1)))) (2)
b(c(x1)) → a(x1) (3)
a(a(c(x1))) → x1 (4)

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(b(c(x1)))) → b#(b(a(a(x1)))) (5)
a#(a(b(c(x1)))) → b#(a(a(x1))) (6)
a#(a(b(c(x1)))) → a#(a(x1)) (7)
a#(a(b(c(x1)))) → a#(x1) (8)
b#(x1) → a#(a(x1)) (9)
b#(x1) → a#(x1) (10)
b#(c(x1)) → a#(x1) (11)

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
-∞ -∞ -∞
-∞ -∞ -∞
· x1
[a(x1)] =
-∞
-∞
-∞
+
0 0 0
0 0 0
-∞ -∞ 0
· x1
[b(x1)] =
0
1
0
+
0 0 0
0 0 0
1 0 0
· x1
[c(x1)] =
0
0
-∞
+
0 0 0
0 0 0
0 0 0
· x1
[b#(x1)] =
0
-∞
-∞
+
-∞ -∞ 0
-∞ -∞ -∞
-∞ -∞ -∞
· x1
the pairs
a#(a(b(c(x1)))) → b#(a(a(x1))) (6)
a#(a(b(c(x1)))) → a#(a(x1)) (7)
a#(a(b(c(x1)))) → a#(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 0
-∞ 0 0
0 0 0
· x1
[b(x1)] =
-∞
-∞
-∞
+
0 0 1
0 0 0
0 0 0
· x1
[c(x1)] =
-∞
-∞
-∞
+
0 0 0
0 0 0
0 0 0
· x1
[b#(x1)] =
-∞
-∞
-∞
+
0 1 0
-∞ -∞ -∞
-∞ -∞ -∞
· x1
the pair
b#(c(x1)) → a#(x1) (11)
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
-∞ -∞ -∞
-∞ -∞ -∞
· x1
[a(x1)] =
0
0
1
+
-∞ 0 0
-∞ -∞ 0
0 -∞ -∞
· x1
[b(x1)] =
1
1
1
+
0 1 1
0 -∞ 0
-∞ 0 0
· x1
[c(x1)] =
-∞
1
0
+
0 0 -∞
0 0 -∞
-∞ -∞ 0
· x1
[b#(x1)] =
1
-∞
-∞
+
0 1 1
-∞ -∞ -∞
-∞ -∞ -∞
· x1
the pair
b#(x1) → a#(x1) (10)
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)] =
-∞
-∞
-∞
+
0 0 0
-∞ -∞ -∞
-∞ -∞ -∞
· x1
[a(x1)] =
-∞
-∞
-∞
+
0 0 -∞
0 0 -∞
0 0 -∞
· x1
[b(x1)] =
-∞
-∞
-∞
+
1 1 1
-∞ -∞ 0
1 1 0
· x1
[c(x1)] =
-∞
-∞
-∞
+
0 0 0
-∞ -∞ -∞
1 1 0
· x1
[b#(x1)] =
-∞
-∞
-∞
+
1 1 0
-∞ -∞ -∞
-∞ -∞ -∞
· x1
the pair
b#(x1) → a#(a(x1)) (9)
could be deleted.

1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

1.1.1.1.1.1.1 P is empty

There are no pairs anymore.