Certification Problem

Input (TPDB SRS_Standard/Waldmann_07_size12/size-12-alpha-3-num-359)

The rewrite relation of the following TRS is considered.

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

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(b(a(x1))) → c(x1) (4)
c(a(x1)) → a(a(b(c(x1)))) (5)

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.
a#(x1) → b#(x1) (6)
b#(b(a(x1))) → c#(x1) (7)
c#(a(x1)) → c#(x1) (8)
c#(a(x1)) → b#(c(x1)) (9)
c#(a(x1)) → a#(b(c(x1))) (10)
c#(a(x1)) → a#(a(b(c(x1)))) (11)

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
-∞ -∞
· x1 +
1 -∞
-∞ -∞
[b(x1)] =
-∞ 0
0 -∞
· x1 +
0 -∞
0 -∞
[a#(x1)] =
-∞ 0
-∞ -∞
· x1 +
1 -∞
-∞ -∞
[c#(x1)] =
0 0
-∞ -∞
· x1 +
-∞ -∞
-∞ -∞
[a(x1)] =
0 0
0 1
· x1 +
0 -∞
2 -∞
[c(x1)] =
-∞ 0
0 1
· x1 +
0 -∞
0 -∞
together with the usable rules
a(x1) → b(x1) (1)
b(b(a(x1))) → c(x1) (4)
c(a(x1)) → a(a(b(c(x1)))) (5)
(w.r.t. the implicit argument filter of the reduction pair), the pair
c#(a(x1)) → a#(b(c(x1))) (10)
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 +
0 -∞
-∞ -∞
[b(x1)] =
0 0
0 -∞
· x1 +
0 -∞
-∞ -∞
[a#(x1)] =
0 1
-∞ -∞
· x1 +
0 -∞
-∞ -∞
[c#(x1)] =
0 1
-∞ -∞
· x1 +
-∞ -∞
-∞ -∞
[a(x1)] =
0 0
0 1
· x1 +
0 -∞
2 -∞
[c(x1)] =
-∞ 0
0 1
· x1 +
1 -∞
1 -∞
together with the usable rules
a(x1) → b(x1) (1)
b(b(a(x1))) → c(x1) (4)
c(a(x1)) → a(a(b(c(x1)))) (5)
(w.r.t. the implicit argument filter of the reduction pair), the pairs
c#(a(x1)) → c#(x1) (8)
c#(a(x1)) → b#(c(x1)) (9)
could be deleted.

1.1.1.1.1 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 2
-∞ -∞
· x1 +
0 -∞
-∞ -∞
[b(x1)] =
-∞ 0
0 0
· x1 +
0 -∞
0 -∞
[a#(x1)] =
0 2
-∞ -∞
· x1 +
0 -∞
-∞ -∞
[c#(x1)] =
2 -∞
-∞ -∞
· x1 +
0 -∞
-∞ -∞
[a(x1)] =
1 0
0 0
· x1 +
1 -∞
0 -∞
[c(x1)] =
1 0
0 -∞
· x1 +
1 -∞
0 -∞
together with the usable rules
a(x1) → b(x1) (1)
b(b(a(x1))) → c(x1) (4)
c(a(x1)) → a(a(b(c(x1)))) (5)
(w.r.t. the implicit argument filter of the reduction pair), the pair
b#(b(a(x1))) → c#(x1) (7)
could be deleted.

1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.