Certification Problem

Input (TPDB SRS_Standard/Waldmann_07_size12/size-12-alpha-3-num-455)

The rewrite relation of the following TRS is considered.

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

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by ttt2 @ termCOMP 2023)

1 Dependency Pair Transformation

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

1.1 Subterm Criterion Processor

We use the projection to multisets
π(b#) = { 1 }
π(c#) = { 1, 1 }
π(a#) = { 1, 1 }
π(b) = { 1 }
π(c) = { 1, 1 }
π(a) = { 1, 1 }
to remove the pairs:
c#(c(x1)) → b#(x1) (7)
c#(c(x1)) → a#(b(x1)) (8)

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
[c#(x1)] =
0 0
-∞ -∞
· x1 +
0 -∞
-∞ -∞
[c(x1)] =
0 2
-1 0
· x1 +
2 -∞
0 -∞
[a#(x1)] =
1 1
-∞ -∞
· x1 +
1 -∞
-∞ -∞
[b#(x1)] =
-∞ 1
-∞ -∞
· x1 +
0 -∞
-∞ -∞
[a(x1)] =
0 1
0 0
· x1 +
1 -∞
0 -∞
[b(x1)] =
-1 1
-2 0
· x1 +
1 -∞
-1 -∞
together with the usable rules
a(x1) → b(c(x1)) (1)
b(a(b(x1))) → c(x1) (2)
c(c(x1)) → a(a(b(x1))) (3)
(w.r.t. the implicit argument filter of the reduction pair), the pair
a#(x1) → c#(x1) (4)
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
[c#(x1)] =
1 0
-∞ -∞
· x1 +
3 -∞
-∞ -∞
[c(x1)] =
0 2
0 -2
· x1 +
2 -∞
-1 -∞
[a#(x1)] =
2 0
-∞ -∞
· x1 +
0 -∞
-∞ -∞
[b#(x1)] =
-2 1
-∞ -∞
· x1 +
0 -∞
-∞ -∞
[a(x1)] =
0 0
2 0
· x1 +
0 -∞
0 -∞
[b(x1)] =
-2 0
-2 0
· x1 +
0 -∞
-∞ -∞
together with the usable rules
a(x1) → b(c(x1)) (1)
b(a(b(x1))) → c(x1) (2)
c(c(x1)) → a(a(b(x1))) (3)
(w.r.t. the implicit argument filter of the reduction pair), the pair
c#(c(x1)) → a#(a(b(x1))) (9)
could be deleted.

1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.