Certification Problem

Input (TPDB SRS_Standard/Waldmann_06_SRS/jw5)

The rewrite relation of the following TRS is considered.

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

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
b(b(a(b(x1)))) → b(a(b(b(b(x1))))) (5)
b(b(a(a(b(x1))))) → b(a(a(b(b(a(b(x1))))))) (6)
b(b(a(a(a(b(x1)))))) → b(a(a(a(b(b(a(a(b(x1))))))))) (7)
b(b(a(a(a(a(b(x1))))))) → b(a(a(a(a(b(b(a(a(a(b(x1))))))))))) (8)

1.1 Semantic Labeling

Root-labeling is applied.

We obtain the labeled TRS
bb(ba(ab(bb(x1)))) → ba(ab(bb(bb(bb(x1))))) (9)
bb(ba(ab(ba(x1)))) → ba(ab(bb(bb(ba(x1))))) (10)
bb(ba(aa(ab(bb(x1))))) → ba(aa(ab(bb(ba(ab(bb(x1))))))) (11)
bb(ba(aa(ab(ba(x1))))) → ba(aa(ab(bb(ba(ab(ba(x1))))))) (12)
bb(ba(aa(aa(ab(bb(x1)))))) → ba(aa(aa(ab(bb(ba(aa(ab(bb(x1))))))))) (13)
bb(ba(aa(aa(ab(ba(x1)))))) → ba(aa(aa(ab(bb(ba(aa(ab(ba(x1))))))))) (14)
bb(ba(aa(aa(aa(ab(bb(x1))))))) → ba(aa(aa(aa(ab(bb(ba(aa(aa(ab(bb(x1))))))))))) (15)
bb(ba(aa(aa(aa(ab(ba(x1))))))) → ba(aa(aa(aa(ab(bb(ba(aa(aa(ab(ba(x1))))))))))) (16)

1.1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.
bb#(ba(ab(bb(x1)))) → bb#(bb(bb(x1))) (17)
bb#(ba(ab(bb(x1)))) → bb#(bb(x1)) (18)
bb#(ba(ab(ba(x1)))) → bb#(bb(ba(x1))) (19)
bb#(ba(ab(ba(x1)))) → bb#(ba(x1)) (20)
bb#(ba(aa(ab(bb(x1))))) → bb#(ba(ab(bb(x1)))) (21)
bb#(ba(aa(ab(ba(x1))))) → bb#(ba(ab(ba(x1)))) (22)
bb#(ba(aa(aa(ab(bb(x1)))))) → bb#(ba(aa(ab(bb(x1))))) (23)
bb#(ba(aa(aa(ab(ba(x1)))))) → bb#(ba(aa(ab(ba(x1))))) (24)
bb#(ba(aa(aa(aa(ab(bb(x1))))))) → bb#(ba(aa(aa(ab(bb(x1)))))) (25)
bb#(ba(aa(aa(aa(ab(ba(x1))))))) → bb#(ba(aa(aa(ab(ba(x1)))))) (26)

1.1.1.1 Reduction Pair Processor

Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals
[bb#(x1)] =
0
-∞
-∞
+
0 0 0
-∞ -∞ -∞
-∞ -∞ -∞
· x1
[ba(x1)] =
0
0
0
+
-∞ 0 -∞
-∞ -∞ 0
-∞ 0 -∞
· x1
[ab(x1)] =
-∞
-∞
-∞
+
0 0 0
-∞ -∞ -∞
0 0 0
· x1
[bb(x1)] =
-∞
-∞
-∞
+
0 0 0
0 0 0
0 0 0
· x1
[aa(x1)] =
-∞
0
0
+
0 1 0
-∞ 0 0
0 0 0
· x1
the pairs
bb#(ba(aa(aa(aa(ab(bb(x1))))))) → bb#(ba(aa(aa(ab(bb(x1)))))) (25)
bb#(ba(aa(aa(aa(ab(ba(x1))))))) → bb#(ba(aa(aa(ab(ba(x1)))))) (26)
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 naturals
[bb#(x1)] =
0
-∞
-∞
+
0 0 0
-∞ -∞ -∞
-∞ -∞ -∞
· x1
[ba(x1)] =
-∞
-∞
-∞
+
0 -∞ -∞
0 -∞ -∞
0 -∞ -∞
· x1
[ab(x1)] =
-∞
-∞
0
+
0 0 0
1 0 0
0 0 0
· x1
[bb(x1)] =
-∞
-∞
-∞
+
0 0 0
-∞ 0 0
0 0 -∞
· x1
[aa(x1)] =
0
-∞
1
+
-∞ -∞ 0
-∞ -∞ -∞
-∞ 0 -∞
· x1
the pairs
bb#(ba(aa(aa(ab(bb(x1)))))) → bb#(ba(aa(ab(bb(x1))))) (23)
bb#(ba(aa(aa(ab(ba(x1)))))) → bb#(ba(aa(ab(ba(x1))))) (24)
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 naturals
[bb#(x1)] =
0
-∞
-∞
+
0 0 0
-∞ -∞ -∞
-∞ -∞ -∞
· x1
[ba(x1)] =
-∞
-∞
0
+
0 0 0
0 0 0
0 0 0
· x1
[ab(x1)] =
-∞
-∞
-∞
+
0 0 -∞
0 0 -∞
0 0 -∞
· x1
[bb(x1)] =
-∞
-∞
0
+
0 0 0
0 0 0
0 0 0
· x1
[aa(x1)] =
0
0
1
+
-∞ -∞ -∞
0 -∞ 1
-∞ -∞ -∞
· x1
the pairs
bb#(ba(aa(ab(bb(x1))))) → bb#(ba(ab(bb(x1)))) (21)
bb#(ba(aa(ab(ba(x1))))) → bb#(ba(ab(ba(x1)))) (22)
could be deleted.

1.1.1.1.1.1.1 Reduction Pair Processor

Using the linear polynomial interpretation over the naturals
[bb#(x1)] = 1 · x1
[ba(x1)] = 1 + 1 · x1
[ab(x1)] = 1 · x1
[bb(x1)] = 1 · x1
[aa(x1)] = 0
the pairs
bb#(ba(ab(bb(x1)))) → bb#(bb(bb(x1))) (17)
bb#(ba(ab(bb(x1)))) → bb#(bb(x1)) (18)
bb#(ba(ab(ba(x1)))) → bb#(bb(ba(x1))) (19)
bb#(ba(ab(ba(x1)))) → bb#(ba(x1)) (20)
could be deleted.

1.1.1.1.1.1.1.1 P is empty

There are no pairs anymore.