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 ttt2 @ termCOMP 2023)

1 Rule Removal

Using the linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1 over the arctic semiring over the integers
[b(x1)] =
1 2
1 0
· x1 +
-∞ -∞
-∞ -∞
[c(x1)] =
0 0
1 0
· x1 +
-∞ -∞
-∞ -∞
[a(x1)] =
0 -∞
0 -∞
· x1 +
-∞ -∞
-∞ -∞
all of the following rules can be deleted.
b(c(x1)) → a(x1) (3)

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.
a#(a(b(c(x1)))) → a#(x1) (5)
a#(a(b(c(x1)))) → a#(a(x1)) (6)
a#(a(b(c(x1)))) → b#(a(a(x1))) (7)
a#(a(b(c(x1)))) → b#(b(a(a(x1)))) (8)
b#(x1) → a#(x1) (9)
b#(x1) → a#(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)] =
0 0
-∞ -∞
· x1 +
0 -∞
-∞ -∞
[b(x1)] =
0 2
0 0
· x1 +
0 -∞
-∞ -∞
[a#(x1)] =
0 0
-∞ -∞
· x1 +
-∞ -∞
-∞ -∞
[c(x1)] =
0 0
0 0
· x1 +
-∞ -∞
-∞ -∞
[a(x1)] =
0 0
0 0
· x1 +
-∞ -∞
-∞ -∞
together with the usable rules
a(a(b(c(x1)))) → b(b(a(a(x1)))) (1)
b(x1) → c(c(a(a(x1)))) (2)
a(a(c(x1))) → x1 (4)
(w.r.t. the implicit argument filter of the reduction pair), the pairs
a#(a(b(c(x1)))) → a#(x1) (5)
a#(a(b(c(x1)))) → a#(a(x1)) (6)
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 +
1 -∞
-∞ -∞
[b(x1)] =
0 2
0 0
· x1 +
1 -∞
1 -∞
[a#(x1)] =
0 0
-∞ -∞
· x1 +
1 -∞
-∞ -∞
[c(x1)] =
0 0
0 0
· x1 +
0 -∞
1 -∞
[a(x1)] =
-∞ 0
0 0
· x1 +
1 -∞
0 -∞
together with the usable rules
a(a(b(c(x1)))) → b(b(a(a(x1)))) (1)
b(x1) → c(c(a(a(x1)))) (2)
a(a(c(x1))) → x1 (4)
(w.r.t. the implicit argument filter of the reduction pair), the pair
a#(a(b(c(x1)))) → b#(a(a(x1))) (7)
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)] =
1 0
-∞ -∞
· x1 +
1 -∞
-∞ -∞
[b(x1)] =
-∞ 0
3 0
· x1 +
0 -∞
0 -∞
[a#(x1)] =
0 -∞
-∞ -∞
· x1 +
0 -∞
-∞ -∞
[c(x1)] =
0 0
0 0
· x1 +
0 -∞
0 -∞
[a(x1)] =
-∞ 0
-∞ 0
· x1 +
0 -∞
0 -∞
together with the usable rules
a(a(b(c(x1)))) → b(b(a(a(x1)))) (1)
b(x1) → c(c(a(a(x1)))) (2)
a(a(c(x1))) → x1 (4)
(w.r.t. the implicit argument filter of the reduction pair), the pair
b#(x1) → a#(x1) (9)
could be deleted.

1.1.1.1.1.1 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 0
-∞ -∞
· x1 +
1 -∞
-∞ -∞
[b(x1)] =
0 1
2 1
· x1 +
1 -∞
2 -∞
[a#(x1)] =
0 0
-∞ -∞
· x1 +
1 -∞
-∞ -∞
[c(x1)] =
0 1
0 0
· x1 +
1 -∞
0 -∞
[a(x1)] =
-∞ 0
-∞ 0
· x1 +
0 -∞
0 -∞
together with the usable rules
a(a(b(c(x1)))) → b(b(a(a(x1)))) (1)
b(x1) → c(c(a(a(x1)))) (2)
a(a(c(x1))) → x1 (4)
(w.r.t. the implicit argument filter of the reduction pair), the pair
a#(a(b(c(x1)))) → b#(b(a(a(x1)))) (8)
could be deleted.

1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.