Certification Problem

Input (TPDB SRS_Standard/Trafo_06/dup11)

The rewrite relation of the following TRS is considered.

a(a(b(b(x1)))) → C(C(x1)) (1)
b(b(c(c(x1)))) → A(A(x1)) (2)
c(c(a(a(x1)))) → B(B(x1)) (3)
A(A(C(C(x1)))) → b(b(x1)) (4)
C(C(B(B(x1)))) → a(a(x1)) (5)
B(B(A(A(x1)))) → c(c(x1)) (6)
a(a(a(a(a(a(a(a(a(a(x1)))))))))) → A(A(A(A(A(A(x1)))))) (7)
A(A(A(A(A(A(A(A(x1)))))))) → a(a(a(a(a(a(a(a(x1)))))))) (8)
b(b(b(b(b(b(b(b(b(b(x1)))))))))) → B(B(B(B(B(B(x1)))))) (9)
B(B(B(B(B(B(B(B(x1)))))))) → b(b(b(b(b(b(b(b(x1)))))))) (10)
c(c(c(c(c(c(c(c(c(c(x1)))))))))) → C(C(C(C(C(C(x1)))))) (11)
C(C(C(C(C(C(C(C(x1)))))))) → c(c(c(c(c(c(c(c(x1)))))))) (12)
B(B(a(a(a(a(a(a(a(a(x1)))))))))) → c(c(A(A(A(A(A(A(x1)))))))) (13)
A(A(A(A(A(A(b(b(x1)))))))) → a(a(a(a(a(a(a(a(C(C(x1)))))))))) (14)
C(C(b(b(b(b(b(b(b(b(x1)))))))))) → a(a(B(B(B(B(B(B(x1)))))))) (15)
B(B(B(B(B(B(c(c(x1)))))))) → b(b(b(b(b(b(b(b(A(A(x1)))))))))) (16)
A(A(c(c(c(c(c(c(c(c(x1)))))))))) → b(b(C(C(C(C(C(C(x1)))))))) (17)
C(C(C(C(C(C(a(a(x1)))))))) → c(c(c(c(c(c(c(c(B(B(x1)))))))))) (18)
a(a(A(A(x1)))) → x1 (19)
A(A(a(a(x1)))) → x1 (20)
b(b(B(B(x1)))) → x1 (21)
B(B(b(b(x1)))) → x1 (22)
c(c(C(C(x1)))) → x1 (23)
C(C(c(c(x1)))) → x1 (24)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by matchbox @ termCOMP 2023)

1 Rule Removal

Using the matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
[C(x1)] = x1 +
1
[B(x1)] = x1 +
1
[A(x1)] = x1 +
1
[c(x1)] = x1 +
2/3
[b(x1)] = x1 +
2/3
[a(x1)] = x1 +
2/3
all of the following rules can be deleted.
a(a(b(b(x1)))) → C(C(x1)) (1)
b(b(c(c(x1)))) → A(A(x1)) (2)
c(c(a(a(x1)))) → B(B(x1)) (3)
A(A(C(C(x1)))) → b(b(x1)) (4)
C(C(B(B(x1)))) → a(a(x1)) (5)
B(B(A(A(x1)))) → c(c(x1)) (6)
a(a(a(a(a(a(a(a(a(a(x1)))))))))) → A(A(A(A(A(A(x1)))))) (7)
A(A(A(A(A(A(A(A(x1)))))))) → a(a(a(a(a(a(a(a(x1)))))))) (8)
b(b(b(b(b(b(b(b(b(b(x1)))))))))) → B(B(B(B(B(B(x1)))))) (9)
B(B(B(B(B(B(B(B(x1)))))))) → b(b(b(b(b(b(b(b(x1)))))))) (10)
c(c(c(c(c(c(c(c(c(c(x1)))))))))) → C(C(C(C(C(C(x1)))))) (11)
C(C(C(C(C(C(C(C(x1)))))))) → c(c(c(c(c(c(c(c(x1)))))))) (12)
a(a(A(A(x1)))) → x1 (19)
A(A(a(a(x1)))) → x1 (20)
b(b(B(B(x1)))) → x1 (21)
B(B(b(b(x1)))) → x1 (22)
c(c(C(C(x1)))) → x1 (23)
C(C(c(c(x1)))) → x1 (24)

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.
C#(C(C(C(C(C(a(a(x1)))))))) → B#(x1) (25)
C#(C(C(C(C(C(a(a(x1)))))))) → B#(B(x1)) (26)
C#(C(b(b(b(b(b(b(b(b(x1)))))))))) → B#(x1) (27)
C#(C(b(b(b(b(b(b(b(b(x1)))))))))) → B#(B(x1)) (28)
C#(C(b(b(b(b(b(b(b(b(x1)))))))))) → B#(B(B(x1))) (29)
C#(C(b(b(b(b(b(b(b(b(x1)))))))))) → B#(B(B(B(x1)))) (30)
C#(C(b(b(b(b(b(b(b(b(x1)))))))))) → B#(B(B(B(B(x1))))) (31)
C#(C(b(b(b(b(b(b(b(b(x1)))))))))) → B#(B(B(B(B(B(x1)))))) (32)
B#(B(B(B(B(B(c(c(x1)))))))) → A#(x1) (33)
B#(B(B(B(B(B(c(c(x1)))))))) → A#(A(x1)) (34)
B#(B(a(a(a(a(a(a(a(a(x1)))))))))) → A#(x1) (35)
B#(B(a(a(a(a(a(a(a(a(x1)))))))))) → A#(A(x1)) (36)
B#(B(a(a(a(a(a(a(a(a(x1)))))))))) → A#(A(A(x1))) (37)
B#(B(a(a(a(a(a(a(a(a(x1)))))))))) → A#(A(A(A(x1)))) (38)
B#(B(a(a(a(a(a(a(a(a(x1)))))))))) → A#(A(A(A(A(x1))))) (39)
B#(B(a(a(a(a(a(a(a(a(x1)))))))))) → A#(A(A(A(A(A(x1)))))) (40)
A#(A(A(A(A(A(b(b(x1)))))))) → C#(x1) (41)
A#(A(A(A(A(A(b(b(x1)))))))) → C#(C(x1)) (42)
A#(A(c(c(c(c(c(c(c(c(x1)))))))))) → C#(x1) (43)
A#(A(c(c(c(c(c(c(c(c(x1)))))))))) → C#(C(x1)) (44)
A#(A(c(c(c(c(c(c(c(c(x1)))))))))) → C#(C(C(x1))) (45)
A#(A(c(c(c(c(c(c(c(c(x1)))))))))) → C#(C(C(C(x1)))) (46)
A#(A(c(c(c(c(c(c(c(c(x1)))))))))) → C#(C(C(C(C(x1))))) (47)
A#(A(c(c(c(c(c(c(c(c(x1)))))))))) → C#(C(C(C(C(C(x1)))))) (48)

1.1.1 Monotonic Reduction Pair Processor with Usable Rules

Using the matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
[C(x1)] = x1 +
1
[B(x1)] = x1 +
1
[A(x1)] = x1 +
1
[c(x1)] = x1 +
2/3
[b(x1)] = x1 +
2/3
[a(x1)] = x1 +
2/3
[C#(x1)] = x1 +
0
[B#(x1)] = x1 +
0
[A#(x1)] = x1 +
0
together with the usable rules
B(B(a(a(a(a(a(a(a(a(x1)))))))))) → c(c(A(A(A(A(A(A(x1)))))))) (13)
A(A(A(A(A(A(b(b(x1)))))))) → a(a(a(a(a(a(a(a(C(C(x1)))))))))) (14)
C(C(b(b(b(b(b(b(b(b(x1)))))))))) → a(a(B(B(B(B(B(B(x1)))))))) (15)
B(B(B(B(B(B(c(c(x1)))))))) → b(b(b(b(b(b(b(b(A(A(x1)))))))))) (16)
A(A(c(c(c(c(c(c(c(c(x1)))))))))) → b(b(C(C(C(C(C(C(x1)))))))) (17)
C(C(C(C(C(C(a(a(x1)))))))) → c(c(c(c(c(c(c(c(B(B(x1)))))))))) (18)
(w.r.t. the implicit argument filter of the reduction pair), the pairs
C#(C(C(C(C(C(a(a(x1)))))))) → B#(x1) (25)
C#(C(C(C(C(C(a(a(x1)))))))) → B#(B(x1)) (26)
C#(C(b(b(b(b(b(b(b(b(x1)))))))))) → B#(x1) (27)
C#(C(b(b(b(b(b(b(b(b(x1)))))))))) → B#(B(x1)) (28)
C#(C(b(b(b(b(b(b(b(b(x1)))))))))) → B#(B(B(x1))) (29)
C#(C(b(b(b(b(b(b(b(b(x1)))))))))) → B#(B(B(B(x1)))) (30)
C#(C(b(b(b(b(b(b(b(b(x1)))))))))) → B#(B(B(B(B(x1))))) (31)
C#(C(b(b(b(b(b(b(b(b(x1)))))))))) → B#(B(B(B(B(B(x1)))))) (32)
B#(B(B(B(B(B(c(c(x1)))))))) → A#(x1) (33)
B#(B(B(B(B(B(c(c(x1)))))))) → A#(A(x1)) (34)
B#(B(a(a(a(a(a(a(a(a(x1)))))))))) → A#(x1) (35)
B#(B(a(a(a(a(a(a(a(a(x1)))))))))) → A#(A(x1)) (36)
B#(B(a(a(a(a(a(a(a(a(x1)))))))))) → A#(A(A(x1))) (37)
B#(B(a(a(a(a(a(a(a(a(x1)))))))))) → A#(A(A(A(x1)))) (38)
B#(B(a(a(a(a(a(a(a(a(x1)))))))))) → A#(A(A(A(A(x1))))) (39)
B#(B(a(a(a(a(a(a(a(a(x1)))))))))) → A#(A(A(A(A(A(x1)))))) (40)
A#(A(A(A(A(A(b(b(x1)))))))) → C#(x1) (41)
A#(A(A(A(A(A(b(b(x1)))))))) → C#(C(x1)) (42)
A#(A(c(c(c(c(c(c(c(c(x1)))))))))) → C#(x1) (43)
A#(A(c(c(c(c(c(c(c(c(x1)))))))))) → C#(C(x1)) (44)
A#(A(c(c(c(c(c(c(c(c(x1)))))))))) → C#(C(C(x1))) (45)
A#(A(c(c(c(c(c(c(c(c(x1)))))))))) → C#(C(C(C(x1)))) (46)
A#(A(c(c(c(c(c(c(c(c(x1)))))))))) → C#(C(C(C(C(x1))))) (47)
A#(A(c(c(c(c(c(c(c(c(x1)))))))))) → C#(C(C(C(C(C(x1)))))) (48)
and no rules could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.