Certification Problem

Input (TPDB SRS_Standard/Waldmann_06_SRS/sym-6)

The rewrite relation of the following TRS is considered.

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

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS
c(c(c(c(x1)))) → b(b(b(b(x1)))) (1)
b(b(x1)) → x1 (2)
b(b(x1)) → c(b(c(x1))) (3)

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.
c#(c(c(c(x1)))) → b#(b(b(b(x1)))) (4)
c#(c(c(c(x1)))) → b#(b(b(x1))) (5)
c#(c(c(c(x1)))) → b#(b(x1)) (6)
c#(c(c(c(x1)))) → b#(x1) (7)
b#(b(x1)) → c#(b(c(x1))) (8)
b#(b(x1)) → b#(c(x1)) (9)
b#(b(x1)) → c#(x1) (10)

1.1.1 Reduction Pair Processor

Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the integers
[c#(x1)] =
0
-∞
-∞
+
-1 -1 -∞
-∞ -∞ -∞
-∞ -∞ -∞
· x1
[c(x1)] =
0
0
-∞
+
-1 0 -1
-1 1 0
-1 -∞ -∞
· x1
[b#(x1)] =
1
-∞
-∞
+
-1 -∞ 0
-∞ -∞ -∞
-∞ -∞ -∞
· x1
[b(x1)] =
0
-∞
0
+
-∞ -∞ 0
-1 -∞ 1
0 -1 -1
· x1
the pair
b#(b(x1)) → c#(b(c(x1))) (8)
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 integers
[c#(x1)] =
0
-∞
-∞
+
-1 -∞ -∞
-∞ -∞ -∞
-∞ -∞ -∞
· x1
[c(x1)] =
0
1
-∞
+
-1 0 -∞
-1 1 0
-1 -∞ -∞
· x1
[b#(x1)] =
0
-∞
-∞
+
-∞ -∞ -1
-∞ -∞ -∞
-∞ -∞ -∞
· x1
[b(x1)] =
0
-1
0
+
-∞ -∞ 0
-1 -∞ 1
0 -1 -∞
· x1
the pairs
c#(c(c(c(x1)))) → b#(b(b(x1))) (5)
c#(c(c(c(x1)))) → 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 integers
[c#(x1)] =
1
-∞
-∞
+
-1 0 0
-∞ -∞ -∞
-∞ -∞ -∞
· x1
[c(x1)] =
0
0
-1
+
-∞ -1 -1
-1 1 -∞
-∞ -1 0
· x1
[b#(x1)] =
-∞
-∞
-∞
+
0 -∞ -∞
-∞ -∞ -∞
-∞ -∞ -∞
· x1
[b(x1)] =
1
-1
-1
+
-1 0 0
0 -∞ -∞
0 -1 -∞
· x1
the pair
b#(b(x1)) → b#(c(x1)) (9)
could be deleted.

1.1.1.1.1.1 Reduction Pair Processor

Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the integers
[c#(x1)] =
1
-∞
-∞
+
0 -1 -1
-∞ -∞ -∞
-∞ -∞ -∞
· x1
[c(x1)] =
0
1
-∞
+
-1 0 -∞
2 -∞ -1
-∞ -1 -1
· x1
[b#(x1)] =
0
-∞
-∞
+
-1 -1 -1
-∞ -∞ -∞
-∞ -∞ -∞
· x1
[b(x1)] =
0
1
2
+
-∞ -∞ -1
-1 -∞ 0
1 0 0
· x1
the pairs
c#(c(c(c(x1)))) → b#(b(b(b(x1)))) (4)
c#(c(c(c(x1)))) → b#(b(x1)) (6)
could be deleted.

1.1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals
[b(x1)] = 1 · x1
[c#(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.

1.1.1.1.1.1.1.1 Size-Change Termination

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

b#(b(x1)) → c#(x1) (10)
1 > 1

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