Certification Problem

Input (TPDB SRS_Standard/Waldmann_07_size12/size-12-alpha-3-num-350)

The rewrite relation of the following TRS is considered.

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

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by ttt2 @ termCOMP 2023)

1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS
a(x1) → b(x1) (1)
b(a(x1)) → c(a(b(x1))) (5)
b(b(x1)) → x1 (3)
c(c(x1)) → a(x1) (4)

1.1 Dependency Pair Transformation

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

1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1 over the arctic semiring over the integers
[b#(x1)] =
1 0
-∞ -∞
· x1 +
-∞ -∞
-∞ -∞
[b(x1)] =
0 -∞
1 0
· x1 +
-∞ -∞
-∞ -∞
[a#(x1)] =
1 0
-∞ -∞
· x1 +
0 -∞
-∞ -∞
[c#(x1)] =
1 -∞
-∞ -∞
· x1 +
0 -∞
-∞ -∞
[a(x1)] =
1 0
1 0
· x1 +
1 -∞
-∞ -∞
[c(x1)] =
0 0
1 0
· x1 +
1 -∞
0 -∞
together with the usable rules
a(x1) → b(x1) (1)
b(a(x1)) → c(a(b(x1))) (5)
b(b(x1)) → x1 (3)
c(c(x1)) → a(x1) (4)
(w.r.t. the implicit argument filter of the reduction pair), the pairs
b#(a(x1)) → b#(x1) (7)
b#(a(x1)) → a#(b(x1)) (8)
could be deleted.

1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1 over the arctic semiring over the integers
[b#(x1)] =
0 0
-∞ -∞
· x1 +
-∞ -∞
-∞ -∞
[b(x1)] =
0 -∞
1 0
· x1 +
-∞ -∞
0 -∞
[a#(x1)] =
0 0
-∞ -∞
· x1 +
0 -∞
-∞ -∞
[c#(x1)] =
0 1
-∞ -∞
· x1 +
0 -∞
-∞ -∞
[a(x1)] =
2 1
1 0
· x1 +
1 -∞
0 -∞
[c(x1)] =
0 1
1 0
· x1 +
0 -∞
1 -∞
together with the usable rules
a(x1) → b(x1) (1)
b(a(x1)) → c(a(b(x1))) (5)
b(b(x1)) → x1 (3)
c(c(x1)) → a(x1) (4)
(w.r.t. the implicit argument filter of the reduction pair), the pair
c#(c(x1)) → a#(x1) (10)
could be deleted.

1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.